Apart from the eighth degree, what do we see here? We recall the 7th grade program. So, remember? This is the formula for abbreviated multiplication, namely the difference of squares! We get:

We carefully look at the denominator. It looks a lot like one of the multipliers in the numerator, but what's wrong? Wrong order of terms. If they were to be reversed, the rule could be applied.

But how to do that? It turns out to be very easy: an even degree of the denominator helps us here.

The terms are magically reversed. This "phenomenon" is applicable to any expression to an even degree: we can freely change the signs in brackets.

But it's important to remember: all signs change at the same time!

Let's go back to the example:

And again the formula:

Whole we call integers, their opposite (that is, taken with the sign "") and a number.

positive integer, but it is no different from natural, then everything looks exactly like in the previous section.

Now let's look at some new cases. Let's start with an indicator equal to.

Any number in the zero degree is equal to one:

As always, let's ask ourselves: why is this so?

Consider a degree with a base. Take, for example, and multiply by:

So, we multiplied the number by, and we got the same as it was -. And what number should you multiply so that nothing changes? That's right, on. Means.

We can do the same with an arbitrary number:

Let's repeat the rule:

Any number in the zero degree is equal to one.

But there are exceptions to many rules. And here it is also there - this is a number (as a base).

On the one hand, it should be equal to any degree - no matter how much you multiply by yourself, you still get zero, this is clear. But on the other hand, like any number in the zero degree, it must equal. So which of this is true? Mathematicians decided not to get involved and refused to raise zero to zero. That is, now we cannot not only divide by zero, but also raise it to a zero power.

Let's go further. In addition to natural numbers and numbers, negative numbers belong to integers. To understand what a negative power is, let's do the same as last time: multiply some normal number by the same negative power:

From here it is already easy to express what you are looking for:

Now we extend the resulting rule to an arbitrary degree:

So, let's formulate a rule:

A number in the negative power is inverse to the same number in the positive power. But at the same time the base cannot be null: (because you cannot divide by).

Let's summarize:

I. Expression not specified in case. If, then.

II. Any number to the zero degree is equal to one:.

III. A number that is not equal to zero is in negative power inverse to the same number in a positive power:.

Tasks for an independent solution:

Well, as usual, examples for an independent solution:

Analysis of tasks for independent solution:

I know, I know, the numbers are terrible, but on the exam you have to be ready for anything! Solve these examples or analyze their solution if you could not solve them and you will learn how to easily cope with them on the exam!

Let's continue to expand the circle of numbers "suitable" as an exponent.

Now consider rational numbers. What numbers are called rational?

Answer: all that can be represented as a fraction, where and are integers, moreover.

To understand what is Fractional degree, consider the fraction:

Let's raise both sides of the equation to the power:

Now let's remember the rule about "Degree to degree":

What number must be raised to a power to get?

This formulation is the definition of the th root.

Let me remind you: the root of the th power of a number () is a number that, when raised to a power, is equal to.

That is, the root of the th power is the inverse of the exponentiation operation:.

Turns out that. Obviously, this particular case can be extended:.

Now we add the numerator: what is it? The answer is easily obtained using the degree-to-degree rule:

But can the base be any number? After all, the root can not be extracted from all numbers.

None!

Remember the rule: any number raised to an even power is a positive number. That is, you cannot extract roots of an even degree from negative numbers!

This means that such numbers cannot be raised to a fractional power with an even denominator, that is, the expression does not make sense.

What about expression?

But here comes the problem.

The number can be represented in the form of other, cancellable fractions, for example, or.

And it turns out that it does exist, but does not exist, but these are just two different records of the same number.

Or another example: once, then you can write. But if we write down the indicator in a different way, and again we get a nuisance: (that is, we got a completely different result!).

To avoid such paradoxes, consider only positive radix with fractional exponent.

So if:

• - natural number;
• - an integer;

Examples:

Rational exponents are very useful for converting rooted expressions, for example:

5 examples to train

Analysis of 5 examples for training

1. Do not forget about the usual properties of degrees:

2.. Here we remember that we forgot to learn the table of degrees:

after all - it is or. The solution is found automatically:.

And now the hardest part. Now we will analyze irrational degree.

All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, with the exception of

After all, by definition irrational numbers - these are numbers that cannot be represented as a fraction, where and are integers (that is, irrational numbers are all real numbers except rational ones).

When studying degrees with a natural, whole and rational indicator, each time we made up a kind of "image", "analogy", or description in more familiar terms.

For example, a natural exponent is a number multiplied several times by itself;

...zero power number - it is, as it were, a number multiplied by itself once, that is, it has not yet begun to be multiplied, which means that the number itself has not even appeared - therefore, the result is only a kind of "blank number", namely the number;

...negative integer degree - it is as if a certain "reverse process" took place, that is, the number was not multiplied by itself, but divided.

By the way, in science, a degree with a complex indicator is often used, that is, the indicator is not even a real number.

But at school we don't think about such difficulties; you will have an opportunity to comprehend these new concepts at the institute.

WHERE WE ARE SURE YOU WILL GO! (if you learn how to solve such examples :))

For instance:

Decide for yourself:

Analysis of solutions:

1. Let's start with the already usual rule for raising a power to a power:

Now look at the indicator. Does he remind you of anything? We recall the formula for reduced multiplication, the difference of squares:

In this case,

Turns out that:

Answer: .

2. We bring fractions in exponents to the same form: either both decimal, or both ordinary. Let's get, for example:

Answer: 16

3. Nothing special, we apply the usual properties of degrees:

ADVANCED LEVEL

Determination of the degree

A degree is an expression of the form:, where:

• — base of degree;
• - exponent.

Degree with a natural exponent (n \u003d 1, 2, 3, ...)

Raising a number to a natural power n means multiplying the number by itself times:

Integer degree (0, ± 1, ± 2, ...)

If the exponent is whole positive number:

Erection to zero degree:

The expression is indefinite, because, on the one hand, in any degree - this, and on the other - any number in the th degree - this.

If the exponent is whole negative number:

(because you cannot divide by).

Once again about zeros: expression is undefined in case. If, then.

Examples:

Rational grade

• - natural number;
• - an integer;

Examples:

Power properties

To make it easier to solve problems, let's try to understand: where did these properties come from? Let's prove them.

Let's see: what is and?

By definition:

So, on the right side of this expression, we get the following product:

But by definition, it is the power of a number with an exponent, that is:

Q.E.D.

Example : Simplify the expression.

Decision : .

Example : Simplify the expression.

Decision : It is important to note that in our rule necessarilymust have the same bases. Therefore, we combine the degrees with the base, but remains a separate factor:

One more important note: this rule is - only for the product of degrees!

By no means should I write that.

Just as with the previous property, let us turn to the definition of the degree:

Let's rearrange this piece like this:

It turns out that the expression is multiplied by itself once, that is, according to the definition, this is the th power of the number:

In essence, this can be called "bracketing the indicator". But you should never do this in total:!

Let's remember the abbreviated multiplication formulas: how many times did we want to write? But this is not true, after all.

A degree with a negative base.

Up to this point, we have only discussed how it should be index degree. But what should be the foundation? In degrees with natural indicator the basis can be any number .

Indeed, we can multiply any numbers by each other, be they positive, negative, or even. Let's think about which signs ("" or "") will have powers of positive and negative numbers?

For example, will the number be positive or negative? AND? ?

With the first, everything is clear: no matter how many positive numbers we multiply by each other, the result will be positive.

But the negative is a little more interesting. After all, we remember a simple rule from the 6th grade: “minus for minus gives plus”. That is, or. But if we multiply by (), we get -.

And so on to infinity: with each subsequent multiplication, the sign will change. You can formulate such simple rules:

1. even degree, - number positive.
2. Negative number raised to odd degree, - number negative.
3. A positive number to any degree is a positive number.
4. Zero to any degree equals zero.

Decide on your own which sign the following expressions will have:

1.	2.	3.
4.	5.	6.

Did you manage? Here are the answers:

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In the first four examples, I hope everything is clear? We just look at the base and exponent and apply the appropriate rule.

In example 5), everything is also not as scary as it seems: it does not matter what the base is equal to - the degree is even, which means that the result will always be positive. Well, except when the base is zero. The foundation is not equal, is it? Obviously not, since (because).

Example 6) is no longer so simple. Here you need to find out which is less: or? If you remember that, it becomes clear that, and therefore, the base is less than zero. That is, we apply rule 2: the result will be negative.

And again we use the definition of degree:

Everything is as usual - we write down the definition of degrees and, divide them into each other, divide them into pairs and get:

Before examining the last rule, let's solve a few examples.

Calculate the values \u200b\u200bof the expressions:

Solutions :

Apart from the eighth degree, what do we see here? We recall the 7th grade program. So, remember? This is the formula for abbreviated multiplication, namely the difference of squares!

We get:

We carefully look at the denominator. It looks a lot like one of the multipliers in the numerator, but what's wrong? Wrong order of terms. If they were swapped, Rule 3 could be applied. But how to do it? It turns out to be very easy: an even degree of the denominator helps us here.

If you multiply it by, nothing changes, right? But now it turns out the following:

The terms are magically reversed. This "phenomenon" is applicable to any expression to an even degree: we can freely change the signs in brackets. But it's important to remember: all signs change simultaneously!It cannot be replaced with by changing only one disadvantage that we do not want!

Let's go back to the example:

And again the formula:

So now the last rule:

How are we going to prove it? Of course, as usual: let's expand the concept of degree and simplify:

Now let's open the brackets. How many letters will there be? times by multipliers - what does it look like? This is nothing more than a definition of an operation multiplication: there were only multipliers. That is, it is, by definition, the degree of a number with an exponent:

Example:

Irrational grade

In addition to the information about the degrees for the intermediate level, we will analyze the degree with an irrational indicator. All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, except - after all, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are whole numbers (that is, irrational numbers are all real numbers except rational).

When studying degrees with a natural, whole and rational indicator, each time we made up a kind of "image", "analogy", or description in more familiar terms. For example, a natural exponent is a number multiplied several times by itself; a number to the zero degree is, as it were, a number multiplied by itself once, that is, it has not yet begun to be multiplied, which means that the number itself has not even appeared - therefore, the result is only a kind of "blank number", namely the number; a degree with an integer negative exponent is as if some kind of "reverse process" took place, that is, the number was not multiplied by itself, but divided.

It is extremely difficult to imagine a degree with an irrational exponent (just as it is difficult to imagine a 4-dimensional space). Rather, it is a purely mathematical object that mathematicians created to extend the concept of a degree to the entire space of numbers.

By the way, in science, a degree with a complex indicator is often used, that is, the indicator is not even a real number. But at school we don't think about such difficulties; you will have an opportunity to comprehend these new concepts at the institute.

So what do we do when we see an irrational exponent? We are trying with all our might to get rid of it! :)

For instance:

Decide for yourself:

1)

2)

3)

Answers:

1. We recall the formula for the difference of squares. Answer:.
2. We bring fractions to the same form: either both decimal, or both ordinary. We get, for example:.
3. Nothing special, we apply the usual degree properties:

SUMMARY OF THE SECTION AND BASIC FORMULAS

Degree is called an expression of the form:, where:

Integer Degree

degree, the exponent of which is a natural number (that is, an integer and positive).

Rational grade

degree, the exponent of which is negative and fractional numbers.

Irrational grade

degree, the exponent of which is an infinite decimal fraction or root.

Power properties

Features of degrees.

• Negative number raised to even degree, - number positive.
• Negative number raised to odd degree, - number negative.
• A positive number to any degree is a positive number.
• Zero is equal to any degree.
• Any number to the zero degree is equal.

NOW YOUR WORD ...

How do you like the article? Write down in the comments if you liked it or not.

Tell us about your experience with degree properties.

Perhaps you have questions. Or suggestions.

Write in the comments.

And good luck with your exams!

We remind you that this lesson understands power properties with natural indicators and zero. Degrees with rational indicators and their properties will be discussed in the lessons for 8 classes.

A natural exponent has several important properties that make it easier to calculate in exponent examples.

Property number 1
Product of degrees

Remember!

When multiplying degrees with the same bases, the base remains unchanged, and the exponents are added.

a m · a n \u003d a m + n, where "a" is any number, and "m", "n" are any natural numbers.

This property of degrees also affects the product of three or more degrees.

• Simplify the expression.
  b b 2 b 3 b 4 b 5 \u003d b 1 + 2 + 3 + 4 + 5 \u003d b 15
• Present as a degree.
  6 15 36 \u003d 6 15 6 2 \u003d 6 15 6 2 \u003d 6 17
• Present as a degree.
  (0.8) 3 (0.8) 12 \u003d (0.8) 3 + 12 \u003d (0.8) 15

Important!

Please note that in the specified property it was only about the multiplication of powers with on the same grounds ... It does not apply to their addition.

You cannot replace the sum (3 3 + 3 2) with 3 5. This is understandable if
count (3 3 + 3 2) \u003d (27 + 9) \u003d 36, and 3 5 \u003d 243

Property number 2
Private degrees

Remember!

When dividing degrees with the same bases, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.

\u003d 11 3 - 2 4 2 - 1 \u003d 11 4 \u003d 44

Example. Solve the equation. We use the property of private degrees.
3 8: t \u003d 3 4

T \u003d 3 8 - 4

Answer: t \u003d 3 4 \u003d 81

Using properties # 1 and # 2, you can easily simplify expressions and perform calculations.

• Example. Simplify the expression.
  4 5m + 6 4 m + 2: 4 4m + 3 \u003d 4 5m + 6 + m + 2: 4 4m + 3 \u003d 4 6m + 8 - 4m - 3 \u003d 4 2m + 5
• Example. Find the value of an expression using the properties of the degree.
  = = = 2 9 + 2

  2 5

  = 2 11

  2 5

  = 2 11 − 5 = 2 6 = 64

  Important!

  Note that property 2 was only about dividing degrees with the same bases.

  The difference (4 3 −4 2) cannot be replaced with 4 1. This is understandable if you count (4 3 −4 2) = (64 − 16) = 48 , and 4 1 \u003d 4

  Be careful!

  Property number 3
  Exponentiation

  Remember!

  When raising a degree to a power, the base of the degree remains unchanged, and the exponents are multiplied.

  (a n) m \u003d a n · m, where "a" is any number, and "m", "n" are any natural numbers.

  
  

  Properties 4
  Degree of work

  Remember!

  When raising to the power of a product, each of the factors is raised to a power. The results are then multiplied.

  (a · b) n \u003d a n · b n, where "a", "b" are any rational numbers; "N" is any natural number.

  • Example 1.
    (6 a 2 b 3 s) 2 \u003d 6 2 a 2 2 b 3 2 s 1 2 \u003d 36 a 4 b 6 s 2
    
  • Example 2.
    (−x 2 y) 6 \u003d ((−1) 6 x 2 6 y 1 6) \u003d x 12 y 6

  Important!

  Note that property # 4, like other degree properties, is applied in reverse.

  (a n b n) \u003d (a b) n

  That is, in order to multiply the powers with the same indicators, you can multiply the bases, and the exponent can be left unchanged.

  • Example. Calculate.
    2 4 5 4 \u003d (2 5) 4 \u003d 10 4 \u003d 10,000
  • Example. Calculate.
    0.5 16 2 16 \u003d (0.5 2) 16 \u003d 1

  In more complex examples there may be cases when multiplication and division must be performed over degrees with different bases and different indicators. In this case, we advise you to proceed as follows.

  For instance, 4 5 3 2 \u003d 4 3 4 2 3 2 \u003d 4 3 (4 3) 2 \u003d 64 12 2 \u003d 64 144 \u003d 9216
  

  An example of raising to a decimal power.

  4 21 (−0.25) 20 \u003d 4 4 20 (−0.25) 20 \u003d 4 (4 (−0.25)) 20 \u003d 4 (−1) 20 \u003d 4 1 \u003d 4

  Properties 5
  Degree of quotient (fraction)

  Remember!

  To raise a quotient to a power, you can raise a separate dividend and a divisor to this power, and divide the first result by the second.

  (a: b) n \u003d a n: b n, where “a”, “b” are any rational numbers, b ≠ 0, n is any natural number.

  • Example. Present the expression in the form of private degrees.
    (5: 3) 12 = 5 12: 3 12

  We remind you that the quotient can be represented as a fraction. Therefore, we dwell on the topic of raising a fraction to a power in more detail on the next page.

Lesson on the topic: "Rules for multiplying and dividing degrees with the same and different exponents. Examples"

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The purpose of the lesson: learn how to perform actions with powers of number.

To begin with, let's remember the concept of "degree of number". An expression like $ \\ underbrace (a * a * \\ ldots * a) _ (n) $ can be represented as $ a ^ n $.

The converse is also true: $ a ^ n \u003d \\ underbrace (a * a * \\ ldots * a) _ (n) $.

This equality is called "notation of the degree as a product". It will help us determine how to multiply and divide degrees.
Remember:
a Is the base of the degree.
n - exponent.
If a n \u003d 1so the number and took once and, accordingly: $ a ^ n \u003d a $.
If a n \u003d 0, then $ a ^ 0 \u003d 1 $.

Why this happens, we can figure out when we get acquainted with the rules of multiplication and division of powers.

Multiplication rules

a) If powers with the same base are multiplied.
To $ a ^ n * a ^ m $, write the degrees as a product: $ \\ underbrace (a * a * \\ ldots * a) _ (n) * \\ underbrace (a * a * \\ ldots * a) _ (m ) $.
The figure shows that the number and have taken n + m times, then $ a ^ n * a ^ m \u003d a ^ (n + m) $.

Example.
$2^3 * 2^2 = 2^5 = 32$.

It is convenient to use this property to simplify the work when raising a number to a large power.
Example.
$2^7= 2^3 * 2^4 = 8 * 16 = 128$.

b) If the powers are multiplied with different bases, but the same exponent.
To $ a ^ n * b ^ n $, write the degrees as a product: $ \\ underbrace (a * a * \\ ldots * a) _ (n) * \\ underbrace (b * b * \\ ldots * b) _ (m ) $.
If we swap the multipliers and count the resulting pairs, we get: $ \\ underbrace ((a * b) * (a * b) * \\ ldots * (a * b)) _ (n) $.

So, $ a ^ n * b ^ n \u003d (a * b) ^ n $.

Example.
$3^2 * 2^2 = (3 * 2)^2 = 6^2= 36$.

Division rules

a) The base of the degree is the same, the indicators are different.
Consider dividing an exponent with a larger exponent by dividing a exponent with a smaller exponent.

So it is necessary $ \\ frac (a ^ n) (a ^ m) $where n\u003e m.

Let's write the powers as a fraction:

$ \\ frac (\\ underbrace (a * a * \\ ldots * a) _ (n)) (\\ underbrace (a * a * \\ ldots * a) _ (m)) $.

For convenience, we will write the division as a simple fraction.

Now let's cancel the fraction.

It turns out: $ \\ underbrace (a * a * \\ ldots * a) _ (n-m) \u003d a ^ (n-m) $.
Hence, $ \\ frac (a ^ n) (a ^ m) \u003d a ^ (n-m) $.

This property will help explain the situation with raising a number to a zero power. Let us assume that n \u003d m, then $ a ^ 0 \u003d a ^ (n-n) \u003d \\ frac (a ^ n) (a ^ n) \u003d 1 $.

Examples.
$ \\ frac (3 ^ 3) (3 ^ 2) \u003d 3 ^ (3-2) \u003d 3 ^ 1 \u003d 3 $.

$ \\ frac (2 ^ 2) (2 ^ 2) \u003d 2 ^ (2-2) \u003d 2 ^ 0 \u003d 1 $.

b) The bases of the degree are different, the indicators are the same.
Let's say you need $ \\ frac (a ^ n) (b ^ n) $. Let's write the powers of numbers as a fraction:

$ \\ frac (\\ underbrace (a * a * \\ ldots * a) _ (n)) (\\ underbrace (b * b * \\ ldots * b) _ (n)) $.

For convenience, let's imagine.

Using the property of fractions, we split the large fraction into the product of small ones, we get.
$ \\ underbrace (\\ frac (a) (b) * \\ frac (a) (b) * \\ ldots * \\ frac (a) (b)) _ (n) $.
Accordingly: $ \\ frac (a ^ n) (b ^ n) \u003d (\\ frac (a) (b)) ^ n $.

Example.
$ \\ frac (4 ^ 3) (2 ^ 3) \u003d (\\ frac (4) (2)) ^ 3 \u003d 2 ^ 3 \u003d 8 $.

Consider the topic of transforming expressions with powers, but first let us dwell on a number of transformations that can be carried out with any expressions, including exponential ones. We will learn how to open parentheses, bring such terms, work with the radix and exponent, use the properties of degrees.

What are exponential expressions?

In the school course, few people use the phrase "exponential expressions", but this term is constantly found in collections to prepare for the exam. In most cases, a phrase denotes expressions that contain degrees in their records. We will reflect this in our definition.

Definition 1

Exponential expression Is an expression that contains degrees.

Here are some examples of exponential expressions, starting with a degree with a natural exponent and ending with a degree with a real exponent.

The simplest power expressions can be considered powers of a number with a natural exponent: 3 2, 7 5 + 1, (2 + 1) 5, (- 0, 1) 4, 2 2 3 3, 3 a 2 - a + a 2, x 3 - 1, (a 2) 3. And also degrees with zero exponent: 5 0, (a + 1) 0, 3 + 5 2 - 3, 2 0. And degrees with integer negative powers: (0, 5) 2 + (0, 5) - 2 2.

It is a little more difficult to work with a degree that has rational and irrational indicators: 264 1 4 - 3 3 3 1 2, 2 3, 5 2 - 2 2 - 1, 5, 1 a 1 4 a 1 2 - 2 a - 1 6 b 1 2, x π x 1 - π, 2 3 3 + 5.

The indicator can be the variable 3 x - 54 - 7 3 x - 58 or the logarithm x 2 l g x - 5 x l g x.

With the question of what power expressions are, we figured out. Now let's get down to converting them.

The main types of transformations of power expressions

First of all, we will look at the basic identity transformations of expressions that can be performed with exponential expressions.

Example 1

Calculate the value of the exponential expression 2 3 (4 2 - 12).

Decision

We will carry out all transformations in compliance with the order of actions. In this case, we will start by performing the actions in brackets: replace the degree with a digital value and calculate the difference between the two numbers. We have 2 3 (4 2 - 12) \u003d 2 3 (16 - 12) \u003d 2 3 4.

It remains for us to replace the degree 2 3 its meaning 8 and calculate the product 8 4 \u003d 32... Here is our answer.

Answer: 2 3 (4 2 - 12) \u003d 32.

Example 2

Simplify the expression with powers 3 a 4 b - 7 - 1 + 2 a 4 b - 7.

Decision

The expression given to us in the problem statement contains similar terms, which we can give: 3 a 4 b - 7 - 1 + 2 a 4 b - 7 \u003d 5 a 4 b - 7 - 1.

Answer: 3 a 4 b - 7 - 1 + 2 a 4 b - 7 \u003d 5 a 4 b - 7 - 1.

Example 3

Present an expression with powers of 9 - b 3 · π - 1 2 as a product.

Decision

Let's represent the number 9 as a power 3 2 and apply the abbreviated multiplication formula:

9 - b 3 π - 1 2 \u003d 3 2 - b 3 π - 1 2 \u003d \u003d 3 - b 3 π - 1 3 + b 3 π - 1

Answer: 9 - b 3 π - 1 2 \u003d 3 - b 3 π - 1 3 + b 3 π - 1.

And now let's move on to the analysis of identical transformations that can be applied precisely in relation to power expressions.

Working with the base and exponent

A degree in the base or exponent can have numbers, variables, and some expressions. For instance, (2 + 0, 3 7) 5 - 3, 7 and ... It is difficult to work with such records. It is much easier to replace an expression in the base of a degree or an expression in an exponent with an identically equal expression.

Conversions of the degree and exponent are carried out according to the rules known to us separately from each other. The most important thing is that as a result of the transformations, an expression identical to the original one is obtained.

The purpose of transformations is to simplify the original expression or obtain a solution to a problem. For example, in the example we gave above, (2 + 0, 3 7) 5 - 3, 7, you can follow the steps to go to the degree 4 , 1 1 , 3 ... Expanding the brackets, we can give similar terms in the base of the degree (a (a + 1) - a 2) 2 (x + 1) and get an exponential expression of a simpler form a 2 (x + 1).

Using degree properties

Power properties, written as equalities, are one of the main tools for transforming power expressions. Here are the main ones, taking into account that a and b Are any positive numbers, and r and s - arbitrary real numbers:

Definition 2

• a r a s \u003d a r + s;
• a r: a s \u003d a r - s;
• (a b) r \u003d a r b r;
• (a: b) r \u003d a r: b r;
• (a r) s \u003d a r s.

In cases where we are dealing with natural, integer, positive exponents, the restrictions on the numbers a and b can be much less strict. So, for example, if we consider the equality a m a n \u003d a m + nwhere m and n Are natural numbers, then it will be true for any values \u200b\u200bof a, both positive and negative, as well as for a \u003d 0.

You can apply the properties of degrees without restrictions in cases where the bases of the degrees are positive or contain variables, the area acceptable values which is such that the bases on it take only positive values. In fact, within school curriculum in mathematics, the student's task is to select a suitable property and apply it correctly.

When preparing for admission to universities, there may be problems in which inaccurate use of properties will lead to a narrowing of the ODZ and other difficulties with the solution. In this section, we will analyze only two such cases. More information on the subject can be found in the topic "Transforming Expressions Using Power Properties".

Example 4

Imagine the expression a 2, 5 (a 2) - 3: a - 5, 5 as a degree with a base a.

Decision

First, we use the exponentiation property and transform the second factor by it (a 2) - 3 ... Then we use the properties of multiplying and dividing powers with the same base:

a 2, 5 a - 6: a - 5, 5 \u003d a 2, 5 - 6: a - 5, 5 \u003d a - 3, 5: a - 5, 5 \u003d a - 3, 5 - (- 5, 5) \u003d a 2.

Answer: a 2, 5 (a 2) - 3: a - 5, 5 \u003d a 2.

Transformation of exponential expressions according to the property of degrees can be performed both from left to right, and in the opposite direction.

Example 5

Find the value of the exponential expression 3 1 3 · 7 1 3 · 21 2 3.

Decision

If we apply equality (a b) r \u003d a r b r, from right to left, then we get a product of the form 3 · 7 1 3 · 21 2 3 and further 21 1 3 · 21 2 3. Let us add the exponents when multiplying degrees with the same bases: 21 1 3 21 2 3 \u003d 21 1 3 + 2 3 \u003d 21 1 \u003d 21.

There is another way to carry out transformations:

3 1 3 7 1 3 21 2 3 \u003d 3 1 3 7 1 3 (3 7) 2 3 \u003d 3 1 3 7 1 3 3 2 3 7 2 3 \u003d \u003d 3 1 3 3 2 3 7 1 3 7 2 3 \u003d 3 1 3 + 2 3 7 1 3 + 2 3 \u003d 3 1 7 1 \u003d 21

Answer: 3 1 3 7 1 3 21 2 3 \u003d 3 1 7 1 \u003d 21

Example 6

A power expression is given a 1, 5 - a 0, 5 - 6, enter new variable t \u003d a 0, 5.

Decision

Imagine the degree a 1, 5 as a 0, 5 3 ... We use the property of degree to degree (a r) s \u003d a r s from right to left and we get (a 0, 5) 3: a 1, 5 - a 0, 5 - 6 \u003d (a 0, 5) 3 - a 0, 5 - 6. You can easily enter a new variable into the resulting expression t \u003d a 0, 5: we get t 3 - t - 6.

Answer: t 3 - t - 6.

Converting fractions containing powers

We usually deal with two variants of exponential expressions with fractions: the expression is a fraction with a power or contains such a fraction. All basic transformations of fractions are applicable to such expressions without restrictions. They can be reduced, reduced to a new denominator, and worked separately with the numerator and denominator. Let us illustrate this with examples.

Example 7

Simplify the exponential expression 3 5 2 3 5 1 3 - 5 - 2 3 1 + 2 x 2 - 3 - 3 x 2.

Decision

We are dealing with a fraction, so we will carry out transformations in both the numerator and the denominator:

3 5 2 3 5 1 3 - 5 - 2 3 1 + 2 x 2 - 3 - 3 x 2 \u003d 3 5 2 3 5 1 3 - 3 5 2 3 5 - 2 3 - 2 - x 2 \u003d \u003d 3 5 2 3 + 1 3 - 3 5 2 3 + - 2 3 - 2 - x 2 \u003d 3 5 1 - 3 5 0 - 2 - x 2

Place a minus in front of the fraction to change the sign of the denominator: 12 - 2 - x 2 \u003d - 12 2 + x 2

Answer: 3 5 2 3 5 1 3 - 5 - 2 3 1 + 2 x 2 - 3 - 3 x 2 \u003d - 12 2 + x 2

Fractions containing powers are reduced to a new denominator in the same way as rational fractions. To do this, you need to find an additional factor and multiply the numerator and denominator of the fraction by it. It is necessary to select an additional factor in such a way that it does not vanish for any values \u200b\u200bof the variables from the ODZ variables for the original expression.

Example 8

Reduce fractions to the new denominator: a) a + 1 a 0, 7 to the denominator a, b) 1 x 2 3 - 2 x 1 3 y 1 6 + 4 y 1 3 to the denominator x + 8 y 1 2.

Decision

a) Let's choose a factor that will allow us to make a reduction to a new denominator. a 0.7 a 0, 3 \u003d a 0.7 + 0, 3 \u003d a,hence, as an additional factor, we take a 0, 3... The range of valid values \u200b\u200bof the variable a includes the set of all positive real numbers. In this area, the degree a 0, 3 does not vanish.

Let's multiply the numerator and denominator of the fraction by a 0, 3:

a + 1 a 0, 7 \u003d a + 1 a 0, 3 a 0, 7 a 0, 3 \u003d a + 1 a 0, 3 a

b) Pay attention to the denominator:

x 2 3 - 2 x 1 3 y 1 6 + 4 y 1 3 \u003d \u003d x 1 3 2 - x 1 3 2 y 1 6 + 2 y 1 6 2

Multiply this expression by x 1 3 + 2 y 1 6, we get the sum of cubes x 1 3 and 2 y 1 6, i.e. x + 8 y 1 2. This is our new denominator, to which we need to reduce the original fraction.

So we found an additional factor x 1 3 + 2 · y 1 6. On the range of admissible values \u200b\u200bof variables x and y the expression x 1 3 + 2 y 1 6 does not vanish, so we can multiply the numerator and denominator of the fraction by it:
1 x 2 3 - 2 x 1 3 y 1 6 + 4 y 1 3 \u003d \u003d x 1 3 + 2 y 1 6 x 1 3 + 2 y 1 6 x 2 3 - 2 x 1 3 y 1 6 + 4 y 1 3 \u003d \u003d x 1 3 + 2 y 1 6 x 1 3 3 + 2 y 1 6 3 \u003d x 1 3 + 2 y 1 6 x + 8 y 1 2

Answer: a) a + 1 a 0, 7 \u003d a + 1 a 0, 3 a, b) 1 x 2 3 - 2 x 1 3 y 1 6 + 4 y 1 3 \u003d x 1 3 + 2 y 1 6 x + 8 y 1 2.

Example 9

Reduce the fraction: a) 30 x 3 (x 0.5 + 1) x + 2 x 1 1 3 - 5 3 45 x 0.5 + 1 2 x + 2 x 1 1 3 - 5 3, b) a 1 4 - b 1 4 a 1 2 - b 1 2.

Decision

a) We use the largest common denominator (GCD), by which the numerator and denominator can be reduced. For numbers 30 and 45, this is 15. We can also reduce by x 0.5 + 1 and on x + 2 x 1 1 3 - 5 3.

We get:

30 x 3 (x 0.5 + 1) x + 2 x 1 1 3 - 5 3 45 x 0.5 + 1 2 x + 2 x 1 1 3 - 5 3 \u003d 2 x 3 3 (x 0.5 + 1)

b) Here, the presence of the same factors is not obvious. You will have to perform some transformations in order to get the same factors in the numerator and denominator. To do this, we expand the denominator using the formula for the difference of squares:

a 1 4 - b 1 4 a 1 2 - b 1 2 \u003d a 1 4 - b 1 4 a 1 4 2 - b 1 2 2 \u003d \u003d a 1 4 - b 1 4 a 1 4 + b 1 4 a 1 4 - b 1 4 \u003d 1 a 1 4 + b 1 4

Answer:a) 30 x 3 (x 0.5 + 1) x + 2 x 1 1 3 - 5 3 45 x 0.5 + 1 2 x + 2 x 1 1 3 - 5 3 \u003d 2 X 3 3 (x 0, 5 + 1), b) a 1 4 - b 1 4 a 1 2 - b 1 2 \u003d 1 a 1 4 + b 1 4.

The main actions with fractions include converting to a new denominator and reducing fractions. Both actions are performed in compliance with a number of rules. When adding and subtracting fractions, first the fractions are reduced to a common denominator, after which actions (addition or subtraction) are performed with the numerators. The denominator remains the same. The result of our actions is a new fraction, the numerator of which is the product of the numerators, and the denominator is the product of the denominators.

Example 10

Follow steps x 1 2 + 1 x 1 2 - 1 - x 1 2 - 1 x 1 2 + 1 1 x 1 2.

Decision

Let's start by subtracting the fractions that are in parentheses. Let's bring them to a common denominator:

x 1 2 - 1 x 1 2 + 1

Subtract the numerators:

x 1 2 + 1 x 1 2 - 1 - x 1 2 - 1 x 1 2 + 1 1 x 1 2 \u003d \u003d x 1 2 + 1 x 1 2 + 1 x 1 2 - 1 x 1 2 + 1 - x 1 2 - 1 x 1 2 - 1 x 1 2 + 1 x 1 2 - 1 1 x 1 2 \u003d \u003d x 1 2 + 1 2 - x 1 2 - 1 2 x 1 2 - 1 x 1 2 + 1 1 x 1 2 \u003d \u003d x 1 2 2 + 2 x 1 2 + 1 - x 1 2 2 - 2 x 1 2 + 1 x 1 2 - 1 x 1 2 + 1 1 x 1 2 \u003d \u003d 4 x 1 2 x 1 2 - 1 x 1 2 + 1 1 x 1 2

Now we multiply the fractions:

4 x 1 2 x 1 2 - 1 x 1 2 + 1 1 x 1 2 \u003d \u003d 4 x 1 2 x 1 2 - 1 x 1 2 + 1 x 1 2

Reduce by the degree x 1 2, we get 4 x 1 2 - 1 x 1 2 + 1.

Additionally, you can simplify the exponential expression in the denominator using the difference of squares: squares formula: 4 x 1 2 - 1 x 1 2 + 1 \u003d 4 x 1 2 2 - 1 2 \u003d 4 x - 1.

Answer: x 1 2 + 1 x 1 2 - 1 - x 1 2 - 1 x 1 2 + 1 1 x 1 2 \u003d 4 x - 1

Example 11

Simplify the exponential expression x 3 4 x 2, 7 + 1 2 x - 5 8 x 2, 7 + 1 3.
Decision

We can reduce the fraction to (x 2, 7 + 1) 2... We get the fraction x 3 4 x - 5 8 x 2, 7 + 1.

Continue converting the degrees of x x 3 4 x - 5 8 · 1 x 2, 7 + 1. Now you can use the property of division of powers with the same bases: x 3 4 x - 5 8 1 x 2, 7 + 1 \u003d x 3 4 - - 5 8 1 x 2, 7 + 1 \u003d x 1 1 8 1 x 2 , 7 + 1.

We pass from the last product to the fraction x 1 3 8 x 2, 7 + 1.

Answer: x 3 4 x 2, 7 + 1 2 x - 5 8 x 2, 7 + 1 3 \u003d x 1 3 8 x 2, 7 + 1.

In most cases, it is more convenient to transfer multipliers with negative exponents from the numerator to the denominator and vice versa, changing the sign of the exponent. This action allows you to simplify the further solution. Here's an example: the exponential expression (x + 1) - 0, 2 3 x - 1 can be replaced by x 3 (x + 1) 0, 2.

Convert expressions with roots and powers

In problems, there are power expressions that contain not only powers with fractional exponents, but also roots. It is desirable to reduce such expressions only to roots or only to degrees. Moving on to degrees is preferable as they are easier to work with. This transition is especially preferable when the LDV of variables for the original expression allows replacing the roots with powers without the need to refer to the module or split the LDV into several intervals.

Example 12

Present the expression x 1 9 x x 3 6 as a power.

Decision

Variable range x is defined by two inequalities x ≥ 0 and x x 3 ≥ 0, which define the set [ 0 , + ∞) .

On this set, we have the right to go from roots to powers:

x 1 9 x x 3 6 \u003d x 1 9 x x x 1 3 1 6

Using the properties of the degrees, we simplify the resulting exponential expression.

x 1 9 x x 1 3 1 6 \u003d x 1 9 x 1 6 x 1 3 1 6 \u003d x 1 9 x 1 6 x 1 1 3 6 \u003d \u003d x 1 9 x 1 6 X 1 18 \u003d x 1 9 + 1 6 + 1 18 \u003d x 1 3

Answer: x 1 9 x x 3 6 \u003d x 1 3.

Converting powers with exponent variables

These transformations are quite simple to carry out if the properties of the degree are used correctly. For instance, 5 2 x + 1 - 3 5 x 7 x - 14 7 2 x - 1 \u003d 0.

We can replace the product of the power, in terms of which is the sum of a variable and a number. On the left side, this can be done with the first and last terms on the left side of the expression:

5 2 x 5 1 - 3 5 x 7 x - 14 7 2 x 7 - 1 \u003d 0.5 5 2 x - 3 5 x 7 x - 2 7 2 x \u003d 0.

Now we divide both sides of the equality by 7 2 x... This expression on the ODZ of the variable x takes only positive values:

5 5 - 3 5 x 7 x - 2 7 2 x 7 2 x \u003d 0 7 2 x, 5 5 2 x 7 2 x - 3 5 x 7 x 7 2 x - 2 7 2 x 7 2 x \u003d 0.5 5 2 x 7 2 x - 3 5 x 7 x 7 x 7 x - 2 7 2 x 7 2 x \u003d 0

Reducing the fractions with powers, we get: 5 5 2 x 7 2 x - 3 5 x 7 x - 2 \u003d 0.

Finally, the ratio of powers with the same exponents is replaced by the powers of the relations, which leads to the equation 5 5 7 2 x - 3 5 7 x - 2 \u003d 0, which is equivalent to 5 5 7 x 2 - 3 5 7 x - 2 \u003d 0.

We introduce a new variable t \u003d 5 7 x, which reduces the solution of the original exponential equation to the solution of the quadratic equation 5 · t 2 - 3 · t - 2 \u003d 0.

Convert expressions with powers and logarithms

Expressions that contain degrees and logarithms are also found in problems. Examples of such expressions are: 1 4 1 - 5 · log 2 3 or log 3 27 9 + 5 (1 - log 3 5) · log 5 3. The transformation of such expressions is carried out using the approaches and properties of logarithms discussed above, which we discussed in detail in the topic "Converting logarithmic expressions".

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Science and Mathematics Articles

Properties of degrees with the same base

There are three properties of degrees with the same bases and natural values. it

• Composition sum
• Private two degrees with the same bases is equal to the expression, where the base is the same, and the exponent is difference indicators of the original factors.
• Raising the power of a number to a power is equal to an expression in which the base is the same number and the exponent is composition two degrees.

Be careful! Rules regarding addition and subtraction degrees with the same bases does not exist.

Let's write these properties-rules in the form of formulas:

• a m? a n \u003d a m + n
• a m? a n \u003d a m – n
• (a m) n \u003d a mn

Now we will consider them with specific examples and try to prove them.

5 2? 5 3 \u003d 5 5 - here we applied the rule; Now let's imagine how we would solve this example if we didn't know the rules:

5 2? 5 3 \u003d 5? five ? five ? five ? 5 \u003d 5 5 - five squared is five times five, and cubed is the product of three fives. The result is the product of five fives, but this is something other than five to the fifth power: 5 5.

3 9? 3 5 \u003d 3 9–5 \u003d 3 4. Let's write the division as a fraction:

It can be shortened:

As a result, we get:

Thus, we have proven that when dividing two degrees with the same bases, their indicators must be subtracted.

However, when dividing, you cannot have the divisor equal to zero (since you cannot divide by zero). In addition, since we consider degrees only with natural exponents, we cannot, as a result of subtracting exponents, obtain a number less than 1. Therefore, the formula a m? a n \u003d a m – n restrictions are imposed: a? 0 and m\u003e n.

Let's move on to the third property:
(2 2) 4 = 2 2?4 = 2 8

Let's write in expanded form:
(2 2) 4 = (2 ? 2) 4 = (2 ? 2) ? (2 ? 2) ? (2 ? 2) ? (2 ? 2) = 2 ? 2 ? 2 ? 2 ? 2 ? 2 ? 2 ? 2 = 2 8

You can come to this conclusion and reasoning logically. You need to multiply two squared four times. But in each square there are two twos, which means there will be eight twos in total.

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Addition and subtraction rules.

1. From a change in the places of the terms, the sum will not change (commutative property of addition)

13 + 25 \u003d 38, can be written as: 25 + 13 \u003d 38

2. The result of addition will not change if adjacent terms are replaced by their sum (associative property of addition).

10 + 13 + 3 + 5 \u003d 31 can be written as: 23 + 3 + 5 \u003d 31; 26 + 5 \u003d 31; 23 + 8 \u003d 31, etc.

3. Units add up to ones, tens to tens, etc.

34 + 11 \u003d 45 (3 tens plus 1 more ten; 4 units plus 1 unit).

4. Units are subtracted from units, tens from tens, etc.

53-12 \u003d 41 (3 units minus 2 units; 5 tens minus 1 tens)

note: 10 units is one dozen. This must be remembered when subtracting, because if the number of units in the deductible is greater than that of the diminished one, then we can "borrow" ten from the diminished one.

41-12 \u003d 29 (In order for 1 to subtract 2, we first have to "borrow" a unit from tens, we get 11-2 \u003d 9; remember that the reduced one has 1 ten less, therefore, there are 3 tens and from him subtract 1 dozen. Answer 29).

5. If one of them is subtracted from the sum of two terms, the second term is obtained.

This means that addition can be verified using subtraction.

To check, one of the terms is subtracted from the sum: 49-7 \u003d 42 or 49-42 \u003d 7

If, as a result of subtraction, you did not receive one of the terms, then there was a mistake in your addition.

6. If you add the subtracted to the difference, you get the subtracted.

This means that subtraction can be verified by addition.

To check the difference, add the subtracted: 19 + 50 \u003d 69.

If, as a result of the procedure described above, you did not receive a reduction, then a mistake was made in your subtraction.

Addition and subtraction of rational numbers

This lesson covers the addition and subtraction of rational numbers. The topic is classified as complex. Here it is necessary to use the entire arsenal of previously acquired knowledge.

The rules for adding and subtracting integers are also valid for rational numbers. Recall that rational numbers are those that can be represented as a fraction, where a - this is the numerator of the fraction, b Is the denominator of the fraction. Moreover b should not be zero.

In this lesson, we will increasingly call fractions and mixed numbers by one general phrase - rational numbers.

Lesson navigation:

Example 1. Find the value of an expression

We enclose each rational number in parentheses along with our signs. We take into account that the plus that is given in the expression is an operation sign and does not apply to a fraction. This fraction has its own plus sign, which is invisible due to the fact that it is not recorded. But we'll write it down for clarity:

This is the addition of rational numbers with different signs... To add rational numbers with different signs, you need to subtract the smaller one from the larger module, and put the sign with the greater module in front of the answer. And in order to understand which module is larger and which is smaller, you need to be able to compare the modules of these fractions before calculating them:

The modulus of a rational number is greater than the modulus of a rational number. Therefore, we subtracted from. We got an answer. Then, having reduced this fraction by 2, we got the final answer.

If desired, some primitive actions, such as enclosing numbers in brackets and affixing modules, can be skipped. This example can be written shorter:

Example 2. Find the value of an expression

We enclose each rational number in parentheses along with our signs. We take into account that the minus that is given in the expression is the sign of the operation and does not apply to the fraction.

The fraction in this case is positive rational numberthat has a plus sign that is invisible. But we'll write it down for clarity:

Let's replace subtraction with addition. Recall that for this you need to add the opposite number to the subtracted one to the one to be reduced:

Received the addition of negative rational numbers. To add negative rational numbers, you need to add their modules and put a minus in front of the answer received:

Example 3. Find the value of an expression

In this expression, fractions have different denominators. To make it easier for ourselves, we bring these fractions to the same (common) denominator. We will not dwell on this in detail. If you're having difficulty, be sure to go back to the fractional lesson and repeat it.

After reducing the fractions to a common denominator, the expression will take the following form:

This is the addition of rational numbers with different signs. We subtract the smaller one from the larger one and put the sign, the module of which is greater, in front of the received answer:

Example 4. Find the value of an expression

We got the sum of three terms. First, we find the value of the expression, then add to the received answer

First action:

Second action:

Thus, the value of the expression is.

The solution for this example can be written shorter

Example 5... Find the value of an expression

Let's put each number in brackets along with our signs. To do this, temporarily expand the mixed number

Let's calculate the whole parts:

In the main expression, instead of write the resulting unit:

Let's collapse the resulting expression. To do this, omit the parentheses and write the unit and the fraction together

The solution for this example can be written shorter:

Example 6. Find the value of an expression

Let's convert the mixed number to an improper fraction. We will rewrite the rest as is:

We enclose each rational number in brackets along with our signs:

Let's replace subtraction with addition:

Received the addition of negative rational numbers. Let's add the modules of these numbers and put a minus in front of the received answer:

Thus, the value of the expression is.

The solution for this example can be written shorter:

Example 7. Find value expression

Let's write down the mixed number in expanded form. Let's rewrite the rest as it is:

We enclose each rational number in brackets together with our own signs

Replace subtraction with addition where possible:

Let's calculate the whole parts:

In the main expression, instead of writing the resulting number? 7

Expression is an expanded form of notation for a mixed number. You can immediately write down the answer by writing together the numbers? 7 and the fraction (hiding the minus of this fraction)

Thus, the value of the expression is

The solution for this example can be written much shorter. If you skip some details, then it can be written as follows:

Example 8. Find the value of an expression

This expression can be evaluated in two ways. Let's consider each of them.

First way. The whole and fractional parts of the expression are evaluated separately.

First, let's write down the mixed numbers in expanded form:

Let's enclose each number in parentheses along with our signs:

Replace subtraction with addition where possible:

We got a sum of several terms. According to the combination law of addition, if the expression contains several terms, then the sum will not depend on the order of actions. This will allow us to group whole and fractional parts separately:

Let's calculate the whole parts:

In the main expression, instead of writing the resulting number? 3

Let's calculate the fractional parts:

In the main expression, instead of writing down the resulting mixed number

To evaluate the resulting expression, the mixed number must be temporarily expanded, then each number must be enclosed in parentheses and the subtraction replaced by addition. This must be done very carefully so as not to confuse the signs of the terms.

After transforming the expression, we got a new expression that is easy to evaluate. A similar expression was in Example 7. Recall that we added the whole parts separately, and left the fractional parts as they are:

So the value of the expression is

The solution for this example can be written shorter

The short solution skips the steps of enclosing numbers in parentheses, replacing subtraction by addition, and adding modules. If you are in school or another educational institution, then you will be required to skip these primitive steps to save time and space. The above short solution can be written even shorter. It will look like this:

Therefore, while at school or in another educational institution, be prepared for the fact that some actions will have to be performed in the mind.

Second way. Mixed numbers of expressions are converted to improper fractions and calculated as ordinary fractions.

Let's put in brackets each rational number together with its signs

Let's replace subtraction with addition:

Now we will mix numbers and convert to improper fractions:

Received the addition of negative rational numbers. Let's add their modules and put a minus in front of the received answer:

Received an answer as last time.

A detailed solution in the second way looks like this:

Example 9. Find Expression Expressions

First way. Let's add the whole and fractional parts separately.

This time, let's try to skip some primitive actions, such as writing an expression in expanded form, enclosing numbers in parentheses, replacing subtraction by addition, adding modules:

Please note that the fractional parts have been brought to a common denominator.

Second way. Let's convert the mixed numbers to improper fractions and calculate them like ordinary fractions.

Example 10. Find the value of an expression

Let's replace subtraction with addition:

In the resulting expression, there are no negative numbers, which are the main reason for making mistakes. And since there are no negative numbers, we can remove the plus in front of the subtracted, and also remove the parentheses. Then we get the simplest expression that can be easily calculated:

In this example, whole and fractional parts were calculated separately.

Example 11. Find the value of an expression

This is the addition of rational numbers with different signs. Let us subtract the smaller one from the larger one and put the sign, the modulus of which is greater, in front of the resulting numbers:

Example 12. Find the value of an expression

The expression consists of several parameters. According to the order of actions, the first step is to perform the actions in brackets.

First we calculate the expression, then the expression. We add the received answers.

First action:

Second action:

Third action:

Answer: expression value equally

Example 13. Find the value of an expression

Let's replace subtraction with addition:

Received by adding rational numbers with different signs. Let us subtract the smaller one from the larger one and put the sign with the greater module in front of the answer. But we are dealing with mixed numbers. To understand which module is larger and which is smaller, you need to compare the modules of these mixed numbers. And in order to compare the modules of mixed numbers, you need to convert them to improper fractions and compare them like ordinary fractions.

The following figure shows all the steps involved in comparing modules of mixed numbers

Having found out which module is larger and which is smaller, we can continue calculating our example:

Thus, the meaning of the expression equally

Consider addition and subtraction of decimal fractions, which also refer to rational numbers and which can be either positive or negative.

Example 14. Find the value of the expression? 3.2 + 4.3

We enclose each rational number in brackets along with our signs. We take into account that the plus that is given in the expression is the sign of the operation and does not apply to the decimal fraction 4.3. This decimal fraction has its own plus sign, which is invisible due to the fact that it is not recorded. But we will write it down for clarity:

This is the addition of rational numbers with different signs. To add rational numbers with different signs, you need to subtract the smaller one from the larger module, and put the sign with the greater module in front of the answer. And in order to understand which module is greater and which is less, you need to be able to compare the modules of these decimal fractions before calculating them:

The absolute value of 4.3 is greater than the absolute value of? 3.2, so we subtract 3.2 from 4.3. The answer was 1.1. The answer is yes, since the answer must contain the sign of the larger module, that is, the module | +4,3 |.

Thus, the value of the expression? 3.2 + (+4.3) is 1.1

Example 15. Find the value of the expression 3.5 + (? 8.3)

This is the addition of rational numbers with different signs. As in the previous example, subtract the smaller one from the larger module and put the sign with the greater module in front of the answer.

3,5 + (?8,3) = ?(|?8,3| ? |3,5|) = ?(8,3 ? 3,5) = ?(4,8) = ?4,8

Thus, the value of the expression 3.5 + (? 8.3) is? 4.8

This example can be written shorter:

Example 16. Find the value of the expression? 7,2 + (? 3,11)

This is the addition of negative rational numbers. To add negative rational numbers, you need to add their modules and put a minus in front of the answer. You can skip the entry with modules so as not to clutter up the expression:

7,2 + (?3,11) = ?7,20 + (?3,11) = ?(7,20 + 3,11) = ?(10,31) = ?10,31

Thus, the value of the expression? 7.2 + (? 3.11) is equal to? 10.31

This example can be written shorter:

Example 17. Find the value of the expression? 0.48 + (? 2.7)

This is the addition of negative rational numbers. Let's add their modules and put a minus sign in front of the received answer. You can skip the entry with modules so as not to clutter up the expression:

0,48 + (?2,7) = (?0,48) + (?2,70) = ?(0,48 + 2,70) = ?(3,18) = ?3,18

Example 18. Find the value of the expression? 4.9? 5.9

We enclose each rational number in parentheses along with our signs. We take into account that the minus that is given in the expression is the sign of the operation and does not apply to the decimal fraction 5.9. This decimal fraction has its own plus sign, which is invisible due to the fact that it is not recorded. But we'll write it down for clarity:

Let's replace subtraction with addition:

Received the addition of negative rational numbers. Add their modules and put a minus in front of the received answer. You can skip the entry with modules so as not to clutter up the expression:

(?4,9) + (?5,9) = ?(4,9 + 5,9) = ?(10,8) = ?10,8

Thus, the value of the expression? 4.9? 5.9 is equal to? 10.8

= ?(4,9 + 5,9) = ?(10,8) = ?10,8

Example 19. Find the value of expression 7? 9.3

Let's put in brackets each number together with its signs

Replace subtraction with addition

Received the addition of rational numbers with different signs. Let us subtract the smaller one from the larger one and put the sign with the greater module in front of the answer. You can skip the entry with modules so as not to clutter up the expression:

(+7) + (?9,3) = ?(9,3 ? 7) = ?(2,3) = ?2,3

So the value of expression 7? 9.3 equals? 2.3

A detailed solution to this example is written as follows:

7 ? 9,3 = (+7) ? (+9,3) = (+7) + (?9,3) = ?(|?9,3| ? |+7|) =

A short solution would look like this:

Example 20. Find the value of the expression? 0.25? (? 1,2)

Let's replace subtraction with addition:

Received the addition of rational numbers with different signs. Let us subtract the smaller one from the larger one and put the sign with the greater module in front of the answer:

0,25 + (+1,2) = |+1,2| ? |?0,25| = 1,2 ? 0,25 = 0,95

A detailed solution to this example is written as follows:

0,25 ? (?1,2) = (?0,25) + (+1,2) = |+1,2| ? |?0,25| = 1,2 ? 0,25 = 0,95

A short solution would look like this:

Example 21. Find the value of the expression? 3.5 + (4.1? 7.1)

First of all, we perform the actions in parentheses, then add the answer received with the number? 3.5. Let's skip the record with modules in order not to clutter up the expressions.

First action:

4,1 ? 7,1 = (+4,1) ? (+7,1) = (+4,1) + (?7,1) = ?(7,1 ? 4,1) = ?(3,0) = ?3,0

Second action:

3,5 + (?3,0) = ?(3,5 + 3,0) = ?(6,5) = ?6,5

Answer: the value of the expression? 3.5 + (4.1? 7.1) is? 6.5.

3,5 + (4,1 ? 7,1) = ?3,5 + (?3,0) = ?6,5

Example 22. Find the value of the expression (3.5? 2.9)? (3.7 x 9.1)

Let's perform the actions in brackets, then subtract the number that was obtained as a result of the second brackets from the number that was obtained as a result of the execution of the first brackets. Let's skip the record with modules in order not to clutter up the expressions.

First action:

3,5 ? 2,9 = (+3,5) ? (+2,9) = (+3,5) + (?2,9) = 3,5 ? 2,9 = 0,6

Second action:

3,7 ? 9,1 = (+3,7) ? (+9,1) = (+3,7) + (?9,1) = ?(9,1 ? 3,7) = ?(5,4) = ?5,4

Third action

0,6 ? (?5,4) = (+0,6) + (+5,4) = 0,6 + 5,4 = 6,0 = 6

Answer: the value of the expression (3.5? 2.9)? (3.7 × 9.1) is 6.

A short solution to this example can be written as follows:

(3,5 ? 2,9) ? (3,7 ? 9,1) = 0,6 ? (?5,4) = 6,0 = 6

Example 23. Find the value of the expression? 3.8 + 17.15? 6.2? 6.15

Let's put in brackets each rational number together with its signs

Replace subtraction with addition where possible

The expression consists of several terms. According to the combination law of addition, if the expression consists of several terms, then the sum will not depend on the order of actions. This means that the terms can be added in any order.

We will not reinvent the wheel, but add all the terms from left to right in their order:

First action:

(?3,8) + (+17,15) = 17,15 ? 3,80 = 13,35

Second action:

13,35 + (?6,2) = 13,35 ? ?6,20 = 7,15

Third action:

7,15 + (?6,15) = 7,15 ? 6,15 = 1,00 = 1

Answer: the value of the expression? 3.8 + 17.15? 6.2? 6.15 equals 1.

A short solution to this example can be written as follows:

3,8 + 17,15 ? 6,2 ? 6,15 = 13,35 + (?6,2) ? 6,15 = 7,15 ? 6,15 = 1,00 = 1

Short solutions create fewer problems and confusion, so it is advisable to get used to them.

Example 24. Find the value of an expression

Let's convert a decimal? 1.8 to a mixed number. We will rewrite the rest as it is. If you are having difficulty converting a decimal to a mixed number, be sure to repeat the decimal lesson.

Example 25. Find the value of an expression

Let's replace subtraction with addition. Along the way, let's convert a decimal fraction (? 4.4) into an improper fraction

There are no negative numbers in the resulting expression. And since there are no negative numbers, we can remove the plus in front of the second number and omit the parentheses. Then we get a simple expression for addition, which can be easily solved

Example 26. Find the value of an expression

Let's convert the mixed number to an improper fraction, and the decimal fraction? 0.85 to an ordinary fraction. We get the following expression:

Received the addition of negative rational numbers. Let's add their modules and put a minus sign in front of the received answer. You can skip the entry with modules so as not to clutter up the expression:

Example 27. Find the value of an expression

Let's convert both fractions to irregular fractions. To convert a decimal fraction 2.05 to an improper fraction, you can convert it first to a mixed number, and then to an improper fraction:

After converting both fractions into improper fractions, we get the following expression:

Received the addition of rational numbers with different signs. Let us subtract the smaller one from the larger one and put the sign with the greater module in front of the received answer:

Example 28. Find the value of an expression

Let's replace subtraction with addition. Along the way, let's convert a decimal fraction to an ordinary fraction

Example 29. Find the value of an expression

Let's convert decimal fractions? 0.25 and? 1.25 into fractions, and leave the rest as it is. We get the following expression:

You can first replace subtraction with addition where possible and add the rational numbers one by one. There is a second option: first, add the rational numbers and, and then subtract the rational number from the resulting number. We will use this option.

First action:

Second action:

Answer: expression value is equal to? 2.

Example 30. Find the value of an expression

Let's convert decimal fractions to ordinary ones. Leave the rest as it is

We got a sum of several terms. If the sum consists of several terms, then the expression can be calculated in any order. This follows from the combination law of addition.

Therefore, we can organize the most convenient option for us. First of all, you can add the first and last terms, namely the rational numbers and. These numbers have the same denominators, which means this will free us from the need to bring them to it.

First action:

The resulting number can be added with the second term, namely with a rational number. Rational numbers and have the same denominators in fractional parts, which again is an advantage for us

Second action:

Well, let's add the resulting number? 7 with the last term, namely with a rational number. Conveniently, when calculating this expression, the sevens will disappear, that is, their sum will be zero, since the sum of opposite numbers is zero.

Third action:

Answer: expression value is

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Adding and subtracting integers

In this tutorial we will learn addition and subtraction of integersas well as rules for adding and subtracting them.

Recall that integers are all positive and negative numbers, as well as the number 0. For example, the following numbers are integers:

Positive numbers can be easily added and subtracted, multiplied and divided. Unfortunately, the same cannot be said for negative numbers, which confuse many newbies with their minuses in front of each digit. As practice shows, mistakes made due to negative numbers upset students the most.

Examples of addition and subtraction of integers

The first thing to learn is how to add and subtract whole numbers using the coordinate line. It is not at all necessary to draw a coordinate line. It is enough to imagine it in your thoughts and see where negative numbers are located and where are positive ones.

Consider the simplest expression: 1 + 3. The value of this expression is 4:

This example can be understood using a coordinate line. To do this, from the point where the number 1 is located, you need to move to the right three steps. As a result, we will find ourselves at the point where the number 4 is located. You can see how this happens in the figure:

The plus sign in the expression 1 + 3 tells us that we should move to the right in the direction of increasing numbers.

Example 2. Find the value of expression 1? 3.

The value of this expression is? 2

This example can again be understood with the help of a coordinate line. To do this, from the point where the number 1 is located, you need to move to the left three steps. As a result, we will find ourselves at the point where the negative number? 2 is located. In the figure, you can see how this happens:

Minus sign in expression 1? 3 tells us that we should move to the left in the direction of decreasing numbers.

In general, you need to remember that if addition is carried out, then you need to move to the right in the direction of increase. If subtraction is carried out, then you need to move to the left in the direction of decreasing.

Example 3. Find the value of the expression? 2 + 4

The value of this expression is 2

This example can again be understood with the help of a coordinate line. To do this, from the point where the negative number? 2 is located, you need to move to the right by four steps. As a result, we will find ourselves at the point where the positive number 2 is located.

It can be seen that we have moved from the point where the negative number? 2 is located to the right side by four steps and ended up at the point where the positive number 2 is located.

The plus sign in the expression? 2 + 4 tells us that we should move to the right in the direction of increasing numbers.

Example 4. Find the value of the expression? 1? 3

The value of this expression is? 4

Again, this example can be solved using the coordinate line. To do this, from the point where the negative number? 1 is located, you need to move to the left by three steps. As a result, we will find ourselves at the point where the negative number? 4

It can be seen that we have moved from the point where the negative number? 1 is located to the left side by three steps and ended up at the point where the negative number? 4 is located.

The minus sign in the expression? 1? 3 tells us that we should move to the left in the direction of decreasing numbers.

Example 5. Find the value of the expression? 2 + 2

The value of this expression is 0

This example can be solved using a coordinate line. To do this, from the point where the negative number? 2 is located, you need to move two steps to the right. As a result, we will find ourselves at the point where the number 0 is located

It can be seen that we have moved from the point where the negative number? 2 is located to the right side by two steps and ended up at the point where the number 0 is located.

The plus sign in the expression? 2 + 2 tells us that we should move to the right in the direction of increasing numbers.

Adding and Subtracting Integer Rules

To calculate this or that expression, it is not necessary to imagine a coordinate line every time, and even more so to draw it. It is more convenient to use ready-made rules.

When applying the rules, you need to pay attention to the sign of the operation and the signs of the numbers that need to be added or subtracted. Which rule to apply will depend on this.

Example 1. Find the value of the expression? 2 + 5

Here, a positive number is added to a negative number. In other words, the addition of numbers with different signs is carried out. ? 2 is negative and 5 is positive. For such cases, the following rule is provided:

So, let's see which module is larger:

The modulus of 5 is greater than the modulus of? 2. The rule requires subtracting the smaller one from the larger module. Therefore, we must subtract 2 from 5, and put the sign, the modulus of which is greater, in front of the received answer.

Number 5 has a higher module, so the sign of this number will be in the answer. That is, the answer is yes:

Usually written shorter? 2 + 5 \u003d 3

Example 2. Find the value of expression 3 + (? 2)

Here, as in the previous example, the addition of numbers with different signs is carried out. 3 is a positive number and? 2 is negative. Note that the number? 2 is enclosed in parentheses to make the expression clearer and prettier. This expression is much easier to understand than the expression 3+? 2.

So, let's apply the rule of adding numbers with different signs. As in the previous example, we subtract the smaller module from the larger module and put the sign with the greater module in front of the answer:

3 + (?2) = |3| ? |?2| = 3 ? 2 = 1

The modulus of the number 3 is greater than the modulus of the number? 2, so we subtracted 2 from 3, and in front of the received answer we put the sign of the modulus, which is greater. Number 3 has a greater module, therefore the sign of this number is put in the answer. That is, the answer is yes.

Usually written shorter than 3 + (? 2) \u003d 1

Example 3. Find the value of expression 3? 7

In this expression, the larger is subtracted from the smaller number. For such a case, the following rule is provided:

To subtract more from a smaller number, you need to subtract the smaller from the larger number and put a minus in front of the answer.

There is a slight catch in this expression. Recall that an equal sign (\u003d) is placed between values \u200b\u200band expressions when they are equal.

Expression value 3? 7 how did we know equal? \u200b\u200b4. This means that any transformations that we make in this expression must be equal to?

But we see that in the second stage is the expression 7? 3, which is not equal to? 4.

To remedy this situation, expression 7? 3 must be enclosed in brackets and a minus must be placed before this bracket:

3 ? 7 = ? (7 ? 3) = ? (4) = ?4

In this case, equality will be observed at each stage:

After the expression is evaluated, the parentheses can be removed, which we did.

Therefore, to be more precise, the solution should look like this:

3 ? 7 = ? (7 ? 3) = ? (4) = ? 4

This rule can be written using variables. It will look like this:

a? b \u003d? (b? a)

A large number of brackets and operation signs can complicate the solution of a seemingly very simple problem, so it is more advisable to learn how to write such examples short, for example 3? 7 \u003d? 4.

In fact, addition and subtraction of integers is reduced only to addition. What does this mean? This means that if you want to subtract numbers, this operation can be replaced by addition.

So let's get acquainted with the new rule:

To subtract one number from another means to add such a number to the decreasing one that will be the opposite of the subtracted one.

For example, consider the simplest expression 5? 3. At the initial stages of studying mathematics, we simply put an equal sign and wrote down the answer:

But now we are progressing in learning, so we need to adapt to the new rules. The new rule says that subtracting one number from another means adding the opposite number to the subtracted number.

Let's try to understand this rule using the example of the expression 5 × 3. Decreased in this expression is 5, and subtracted is 3. The rule says that in order to subtract 3 from 5, you need to add to 5 such a number that will be opposite to 3. The opposite for the number 3 is the number? 3. We write a new expression:

We already know how to find values \u200b\u200bfor such expressions. This is the addition of numbers with different signs, which we discussed above. To add numbers with different signs, you need to subtract the smaller one from the larger module, and put the sign with the greater module in front of the answer:

5 + (?3) = |5| ? |?3| = 5 ? 3 = 2

The modulus of number 5 is greater than the modulus of number? 3. Therefore, we subtracted 3 from 5 and got 2. The number 5 has a greater module, so the sign of this number was put in the answer. That is, the answer is yes.

Not everyone is able to quickly replace subtraction with addition at first. This is due to the fact that positive numbers are written without their plus sign.

For example, in expression 3? The 1 minus sign indicating subtraction is an operation sign and does not refer to one. The unit in this case is a positive number and it has its own plus sign, but we do not see it, since the plus in front of positive numbers is traditionally not written.

Therefore, for clarity, this expression can be written as follows:

For convenience, numbers with their own signs are enclosed in parentheses. In this case, replacing subtraction with addition is much easier. The subtracted in this case is the number (+1), and the opposite number (? 1). We replace the operation of subtraction by addition and instead of the subtracted (+1) we write the opposite number (? 1)

(+3) ? (+1) = (+3) + (?1) = |+3| ? |?1| = 3 ? 1 = 2

At first glance, it will seem what is the point in these unnecessary gestures, if you can put an equal sign with the good old method and immediately write down the answer 2. In fact, this rule will help us out more than once.

Let's solve the previous example 3? 7 using the rule of subtraction. First, we bring the expression to its normal form, giving each number its own signs. The three has a plus sign because it is a positive number. The minus indicating subtraction does not apply to 7. The 7 has a plus sign because it is also a positive number:

Let's replace subtraction with addition:

Further calculation is not difficult:

Example 7. Find the value of the expression? 4? five

We have before us again the operation of subtraction. This operation must be replaced by addition. To the reduced (? 4) add the opposite number to the subtracted (+5). The opposite number for the subtracted (+5) is the number (? 5).

We've come to a situation where negative numbers need to be added. For such cases, the following rule is provided:

To add negative numbers, you need to add their modules, and put a minus in front of the received answer.

So, let's add the modules of numbers, as the rule requires of us, and put a minus in front of the received answer:

(?4) ? (+5) = (?4) + (?5) = |?4| + |?5| = 4 + 5 = ?9

The entry with modules must be enclosed in brackets and a minus must be placed before these brackets. This will provide the minus that must come before the answer:

(?4) ? (+5) = (?4) + (?5) = ?(|?4| + |?5|) = ?(4 + 5) = ?(9) = ?9

The solution for this example can be written shorter:

Example 8. Find the value of the expression? 3? five ? 7? nine

Let's bring the expression to an understandable form. Here all numbers except? 3 are positive, so they will have plus signs:

Let's replace the operations of subtraction with operations of addition. All minuses (except for the minus, which is in front of the three) will change to pluses and all positive numbers will change to the opposite:

Now let's apply the rule for adding negative numbers. To add up negative numbers, you need to add their modules and put a minus in front of the answer received:

= ?(|?3| + |?5| + |?7| + |?9|) = ?(3 + 5 + 7 + 9) = ?(24) = ?24

The solution for this example can be written shorter:

3 ? 5 ? 7 ? 9 = ?(3 + 5 + 7 + 9) = ?24

Example 9. Find the value of the expression? 10 + 6? 15 + 11? 7

Let's bring the expression to an understandable form:

There are two operations here at once: addition and subtraction. We leave the addition as it is, and replace the subtraction with addition:

(?10) + (+6) ? (+15) + (+11) ? (+7) = (?10) + (+6) + (?15) + (+11) + (?7)

Observing the order of actions, we will perform each action in turn, relying on the previously learned rules. You can skip entries with modules:

First action:

(?10) + (+6) = ? (10 ? 6) = ? (4) = ? 4

Second action:

(?4) + (?15) = ? (4 + 15) = ? (19) = ? 19

Third action:

(?19) + (+11) = ? (19 ? 11) = ? (8) = ?8

Fourth action:

(?8) + (?7) = ? (8 + 7) = ? (15) = ? 15

Thus, the value of the expression? 10 + 6? 15 + 11? 7 equals? 15

Note... It is not at all necessary to reduce the expression to an understandable form, enclosing numbers in brackets. When getting used to negative numbers, you can skip this step as it takes time and can be confusing.

So, to add and subtract whole numbers, you need to remember the following rules:

To add numbers with different signs, you need to subtract the smaller module from the larger module, and put the sign with the greater module before the answer.

To subtract more from a smaller number, you need to subtract the smaller from the larger number and put a minus sign in front of the answer.

Subtracting one number from another means adding the opposite number to the number to be subtracted.

To add negative numbers, you need to add their modules, and put a minus sign in front of the received answer.

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