Lesson topic: Exponentiation of product, quotient and degree

Lesson type: Lesson of generalization and systematization of knowledge

Formed results:

Subject. To consolidate the skills of applying the properties of a degree with a natural indicator

Personal. To form the ability to plan their actions in accordance with the training task

Metasubject. Develop understanding the essence of algebraic prescriptions and the ability to act in accordance with the proposed algorithm

Expected outcomes: Students will learn how to use the properties of a degree with a natural exponent to calculate the value of expressions and transform expressions containing degrees.

Equipment: cards, multimedia projector, signal cards for reflection.

Organizational structure of the lesson:

1 . Organizing time.

Hello dear guys! I'm very glad to see you. Let's start the math lesson

What were the difficulties in performing the d / s?

Reflection.

In front of each student are circles of three colors: red, green, blue.

Tell me about your mood using colored circles (red– joyful, I am sure that I will learn a lot of new things at the lesson, I am confident in my knowledge.

Green -calm; I am confident in my knowledge.

Blue- anxious; I'm not sure).

I'll cheer you up a bit with Poisson's words: "Life is adorned by two things: doing mathematics and teaching it."

Let's decorate our lives!

2. Communication of the topic and purpose of the lesson.

Today we will continue our study of the topic: “Raising the product of a quotient and a degree to a power”,

consolidate all the studied actions with degrees,

We will learn to reason, think logically and prove our point of view.

3. Blitz poll according to the rules of the topic.

How to multiply exponents with the same base? Give examples.

How to divide degrees with the same base?

What is the power of a non-zero number a with zero exponent?

How to raise a product to a power?

How to raise a degree to a degree?

4. Oral account.

To whom do these words belong?

“Among all the sciences that open the way for man to the knowledge of the laws of nature, the most powerful, the greatest science is mathematics.”

/Sofya Vasilievna Kovalevskaya/

The first woman is a mathematician.

You will learn by completing the tasks of oral counting.

K - What is the side of the square if its area is 49cm 2. (7cm)

O - The square of what number is equal to? ()

B - x 3 x 4 (x 7)

A - x 6 : x 2 (x 4)

L - (x 3) 3 (x 9)

E -
(m 3 )

AT -
(m 8 )

FROM -
(m 10 )

K - (- 2) 3 (-8)

A - - 2 2 (-4)

I - 2 0 (1)

5. Consolidation of the studied.

We repeated the rules for raising a product to a power and a power to a power.

Now let's fix it on practical tasks.

Several people willresearch. (Slide)

Work in pairs.

1) Prove that the squares of opposite numbers are equal.

2) Prove that the cubes of opposite numbers are opposite.

3) How will the area of ​​a square change if its side is doubled? 3 times; 10 times; n times?

4) How will the volume of the cube change if its edge is doubled; 3 times; 10 times; n times?

6. Reflection: show me your mood.

7. Fizminutka: "I agree - I do not agree"

Nod your head if you agree with me or not.

1) (y 2) 3 \u003d y 5 (no)

2) (-3) 3 = -27 (yes)

3) (-x) 2 \u003d -x 2 (no)

4) The graph of the function y \u003d 1.3x passes through the origin. (Yes)

8.

3 · () 2 – 0,5 2

a) -1; b) - 1 ; in 1 ; d) 1

2) Simplify the expression:

a) m 10 ; b)m 4 ; c) m 2 ; d) m 8 .

3) Calculate:

A) 3; b) 9; c) : d)

4) What expression should be substituted instead of (*) to get the identity:

X 8 : (*) = x 4

A) x 4; b) x 2; c) x 8; d) x 12

Checking the slide test:

9. Let's play "Find the mistake!"

1) a 15 : a 3 = a 5

2) -z · z5 · z 0 =-z 6 - right

3)
=

4) (y 4 y) 2 \u003d y 10 - true

Write down the incorrect tasks and solve correctly.

10. The result of the lesson.

What did you learn in the lesson?

11. D / s

No. 458, 457 (slide)

Reports about S.V. Kovalevskaya.

12. Reflection.

Show how you feel when you leave the class.

Slide: Good luck!

FI:

Independent work. (test)

1) Find the value of the expression:

3 () 2 – 0.5 2

a) -1; b) - 1 ; in 1 ; d) 1

2) Simplify the expression:

a) m 10 ; b)m 4 ; c) m 2 ; d) m 8 .

3) Calculate:

a) 3; b) 9; c) : d)

4) What expression should be substituted instead of (*) to get the identity:

x 8 : (*) = x 4

a) x 4; b) x 2; c) x 8; d) x 12

Grade:

Independent work. (test)

1) Find the value of the expression:

3 () 2 – 0.5 2

a) -1; b) - 1 ; in 1 ; d) 1

2) Simplify the expression:

If we do not pay attention to the eighth degree, what do we see here? Let's take a look at the 7th grade program. So, remember? This is the abbreviated multiplication formula, namely the difference of squares! We get:

We carefully look at the denominator. It looks a lot like one of the numerator factors, but what's wrong? Wrong order of terms. If they were swapped, the rule could apply.

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

The terms have magically changed places. This "phenomenon" applies 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 name the natural numbers, their opposites (that is, taken with the sign "") and the number.

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

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

Any number to the zero power is equal to one:

As always, we ask ourselves: why is this so?

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

So, we multiplied the number by, and got the same as it was -. What number must be multiplied by 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 to the zero power 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 must be equal to any degree - no matter how much you multiply zero by itself, you still get zero, this is clear. But on the other hand, like any number to the zero degree, it must be equal. So what is the truth of this? Mathematicians decided not to get involved and refused to raise zero to the zero power. That is, now we can not only divide by zero, but also raise it to the zero power.

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

From here it is already easy to express the desired:

Now we extend the resulting rule to an arbitrary degree:

So, let's formulate the rule:

A number to a negative power is the inverse of the same number to a positive power. But at the same time base cannot be null:(because it is impossible to divide).

Let's summarize:

I. Expression is not defined in case. If, then.

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

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

Tasks for independent solution:

Well, as usual, examples for an independent solution:

Analysis of tasks for independent solution:

I know, I know, the numbers are scary, but at the exam you have to be ready for anything! Solve these examples or analyze their solution if you couldn't solve it and you will learn how to easily deal with them in the exam!

Let's continue to expand the range 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" Let's consider a fraction:

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

Now remember the rule "degree to degree":

What number must be raised to a power to get?

This formulation is the definition of the root of the th degree.

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

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

It turns out that. Obviously, this special case can be extended: .

Now add the numerator: what is it? The answer is easy to get with the power-to-power 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, it is impossible to extract roots of an even degree from negative numbers!

And 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 a problem arises.

The number can be represented as other, reduced fractions, for example, or.

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

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

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

So if:

• - natural number;
• is an integer;

Examples:

Powers with a rational exponent are very useful for transforming expressions with roots, for example:

5 practice examples

Analysis of 5 examples for training

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

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

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

Well, now - the most difficult. Now we will analyze degree with an irrational exponent.

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

Indeed, by definition, irrational numbers 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, integer and rational indicator, each time we made up a certain “image”, “analogy”, or description in more familiar terms.

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

...zero power- this 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 yet - therefore, the result is only a certain “preparation of a number”, namely a number;

...negative integer exponent- it’s as if a certain “reverse process” has taken place, that is, the number was not multiplied by itself, but divided.

By the way, science often uses a degree with a complex exponent, that is, an exponent is not even a real number.

But at school, we don’t think about such difficulties; you will have the 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 example:

Decide for yourself:

Analysis of solutions:

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

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

In this case,

It turns out that:

Answer: .

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

Answer: 16

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

ADVANCED LEVEL

Definition of degree

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

• — base of degree;
• - exponent.

Degree with natural exponent (n = 1, 2, 3,...)

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

Power with integer exponent (0, ±1, ±2,...)

If the exponent is positive integer number:

erection to zero power:

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

If the exponent is integer negative number:

(because it is impossible to divide).

One more time about nulls: the expression is not defined in the case. If, then.

Examples:

Degree with rational exponent

• - natural number;
• is an integer;

Examples:

Degree 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, the following product is obtained:

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

Q.E.D.

Example : Simplify the expression.

Solution : .

Example : Simplify the expression.

Solution : It is important to note that in our rule necessarily must be on the same basis. Therefore, we combine the degrees with the base, but remain a separate factor:

Another important note: this rule - only for products of powers!

Under no circumstances should I write that.

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

Let's rearrange it 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 fact, this can be called "bracketing the indicator". But you can never do this in total:!

Let's recall the formulas for abbreviated multiplication: how many times did we want to write? But that's not true, really.

Power with a negative base.

Up to this point, we have discussed only what should be index degree. But what should be the basis? In degrees from natural indicator the basis may be any number .

Indeed, we can multiply any number by each other, whether they are positive, negative, or even. Let's think about what signs ("" or "") will have degrees of positive and negative numbers?

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

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

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

And so on ad infinitum: with each subsequent multiplication, the sign will change. You can formulate these simple rules:

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

Determine for yourself what 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 simply 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 base is not the same, 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, which means that 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 analyzing the last rule, let's solve a few examples.

Calculate the values ​​of expressions:

Solutions :

If we do not pay attention to the eighth degree, what do we see here? Let's take a look at the 7th grade program. So, remember? This is the abbreviated multiplication formula, namely the difference of squares!

We get:

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

If you multiply it by, nothing changes, right? But now it looks like this:

The terms have magically changed places. This "phenomenon" applies 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! It cannot be replaced by by changing only one objectionable minus to us!

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:

Well, now let's open the brackets. How many letters will there be? times by multipliers - what does it look like? This is nothing but the definition of an operation multiplication: total there turned out to be multipliers. That is, it is, by definition, a power of a number with an exponent:

Example:

Degree with irrational exponent

In addition to information about the degrees for the average 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, with the exception - after all, by definition, irrational numbers 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, integer and rational indicator, each time we made up a certain “image”, “analogy”, or description in more familiar terms. For example, a natural exponent is a number multiplied by itself several times; 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 yet - therefore, the result is only a certain “preparation of a number”, namely a number; a degree with an integer negative indicator - it is as if a certain “reverse process” has occurred, 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 have created to extend the concept of a degree to the entire space of numbers.

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

So what do we do if we see an irrational exponent? We are trying our best to get rid of it! :)

For example:

Decide for yourself:

1)

2)

3)

Answers:

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

SECTION SUMMARY AND BASIC FORMULA

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

Degree with integer exponent

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

Degree with rational exponent

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

Degree with irrational exponent

exponent whose exponent is an infinite decimal fraction or root.

Degree 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 power is a positive number.
• Zero is equal to any power.
• Any number to the zero power is equal.

NOW YOU HAVE A WORD...

How do you like the article? Let me know in the comments below if you liked it or not.

Tell us about your experience with the power properties.

Perhaps you have questions. Or suggestions.

Write in the comments.

And good luck with your exams!

Exponentiation is an operation closely related to multiplication, this operation is the result of multiple multiplication of a number by itself. Let's represent the formula: a1 * a2 * ... * an = an.

For example, a=2, n=3: 2 * 2 * 2=2^3 = 8 .

In general, exponentiation is often used in various formulas in mathematics and physics. This function has a more scientific purpose than the four basic ones: Addition, Subtraction, Multiplication, Division.

Raising a number to a power

Raising a number to a power is not a difficult operation. It is related to multiplication like the relationship between multiplication and addition. Record an - a short record of the n-th number of numbers "a" multiplied by each other.

Consider exponentiation on the simplest examples, moving on to complex ones.

For example, 42. 42 = 4 * 4 = 16 . Four squared (to the second power) equals sixteen. If you do not understand the multiplication 4 * 4, then read our article about multiplication.

Let's look at another example: 5^3. 5^3 = 5 * 5 * 5 = 25 * 5 = 125 . Five cubed (to the third power) equals one hundred and twenty-five.

Another example: 9^3. 9^3 = 9 * 9 * 9 = 81 * 9 = 729 . Nine cubed equals seven hundred twenty-nine.

Exponentiation Formulas

To correctly raise to a power, you need to remember and know the formulas below. There is nothing beyond natural in this, the main thing is to understand the essence and then they will not only be remembered, but also seem easy.

Raising a monomial to a power

What is a monomial? This is the product of numbers and variables in any quantity. For example, two is a monomial. And this article is about raising such monomials to a power.

Using exponentiation formulas, it will not be difficult to calculate the exponentiation of a monomial to a power.

For example, (3x^2y^3)^2= 3^2 * x^2 * 2 * y^(3 * 2) = 9x^4y^6; If you raise a monomial to a power, then each component of the monomial is raised to a power.

When raising a variable that already has a degree to a power, the degrees are multiplied. For example, (x^2)^3 = x^(2 * 3) = x^6 ;

Raising to a negative power

A negative exponent is the reciprocal of a number. What is a reciprocal? For any number X, the reciprocal is 1/X. That is X-1=1/X. This is the essence of the negative degree.

Consider the example (3Y)^-3:

(3Y)^-3 = 1/(27Y^3).

Why is that? Since there is a minus in the degree, we simply transfer this expression to the denominator, and then raise it to the third power. Just right?

Raising to a fractional power

Let's start with a specific example. 43/2. What does power 3/2 mean? 3 - numerator, means raising a number (in this case 4) to a cube. The number 2 is the denominator, this is the extraction of the second root of the number (in this case 4).

Then we get the square root of 43 = 2^3 = 8 . Answer: 8.

So, the denominator of a fractional degree can be either 3 or 4, and to infinity any number, and this number determines the degree of the square root extracted from a given number. Of course, the denominator cannot be zero.

Raising a root to a power

If the root is raised to a power equal to the power of the root itself, then the answer is the radical expression. For example, (√x)2 = x. And so in any case of equality of the degree of the root and the degree of raising the root.

If (√x)^4. Then (√x)^4=x^2. To check the solution, we translate the expression into an expression with a fractional degree. Since the root is square, the denominator is 2. And if the root is raised to the fourth power, then the numerator is 4. We get 4/2=2. Answer: x = 2.

In any case, the best option is to simply convert the expression to a fractional exponent. If the fraction is not reduced, then such an answer will be, provided that the root of the given number is not allocated.

Exponentiation of a complex number

What is a complex number? A complex number is an expression that has the formula a + b * i; a, b are real numbers. i is the number that, when squared, gives the number -1.

Consider an example. (2 + 3i)^2.

(2 + 3i)^2 = 22 +2 * 2 * 3i +(3i)^2 = 4+12i^-9=-5+12i.

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Exponentiation online

With the help of our calculator, you can calculate the exponentiation of a number to a power:

Exponentiation Grade 7

Raising to a power begins to pass schoolchildren only in the seventh grade.

Exponentiation is an operation closely related to multiplication, this operation is the result of multiple multiplication of a number by itself. Let's represent the formula: a1 * a2 * … * an=an .

For example, a=2, n=3: 2 * 2 * 2 = 2^3 = 8.

Solution Examples:

Exponentiation presentation

Presentation on exponentiation, designed for seventh graders. The presentation may clarify some incomprehensible points, but there will probably not be such points thanks to our article.

Outcome

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In continuation of the conversation about the degree of a number, it is logical to deal with finding the value of the degree. This process has been named exponentiation. In this article, we will just study how exponentiation is performed, while we will touch on all possible exponents - natural, integer, rational and irrational. And by tradition, we will consider in detail the solutions to examples of raising numbers to various degrees.

Page navigation.

What does "exponentiation" mean?

Let's start by explaining what is called exponentiation. Here is the relevant definition.

Definition.

Exponentiation is to find the value of the power of a number.

Thus, finding the value of the power of a with the exponent r and raising the number a to the power of r is the same thing. For example, if the task is “calculate the value of the power (0.5) 5”, then it can be reformulated as follows: “Raise the number 0.5 to the power of 5”.

Now you can go directly to the rules by which exponentiation is performed.

Raising a number to a natural power

In practice, equality based on is usually applied in the form . That is, when raising the number a to a fractional power m / n, the root of the nth degree from the number a is first extracted, after which the result is raised to an integer power m.

Consider solutions to examples of raising to a fractional power.

Example.

Calculate the value of the degree.

Solution.

We show two solutions.

First way. By definition of degree with a fractional exponent. We calculate the value of the degree under the sign of the root, after which we extract the cube root: .

The second way. By definition of a degree with a fractional exponent and on the basis of the properties of the roots, the equalities are true . Now extract the root Finally, we raise to an integer power .

Obviously, the obtained results of raising to a fractional power coincide.

Answer:

Note that the fractional exponent can be written as a decimal fraction or a mixed number, in these cases it should be replaced by the corresponding ordinary fraction, and then exponentiation should be performed.

Example.

Calculate (44.89) 2.5 .

Solution.

We write the exponent in the form of an ordinary fraction (if necessary, see the article): . Now we perform raising to a fractional power:

Answer:

(44,89) 2,5 =13 501,25107 .

It should also be said that raising numbers to rational powers is a rather laborious process (especially when the numerator and denominator of the fractional exponent are quite large numbers), which is usually carried out using computer technology.

In conclusion of this paragraph, we will dwell on the construction of the number zero to a fractional power. We gave the following meaning to the fractional degree of zero of the form: for we have , while zero to the power m/n is not defined. So, zero to a positive fractional power is zero, for example, . And zero in a fractional negative power does not make sense, for example, the expressions and 0 -4.3 do not make sense.

Raising to an irrational power

Sometimes it becomes necessary to find out the value of the degree of a number with an irrational exponent. In this case, for practical purposes, it is usually sufficient to obtain the value of the degree up to a certain sign. We note right away that in practice this value is calculated using electronic computing technology, since manual raising to an irrational power requires a large number of cumbersome calculations. But nevertheless we will describe in general terms the essence of the actions.

To get an approximate value of the exponent of a with an irrational exponent, some decimal approximation of the exponent is taken, and the value of the exponent is calculated. This value is the approximate value of the degree of the number a with an irrational exponent. The more accurate the decimal approximation of the number is taken initially, the more accurate the degree value will be in the end.

As an example, let's calculate the approximate value of the power of 2 1.174367... . Let's take the following decimal approximation of an irrational indicator: . Now we raise 2 to a rational power of 1.17 (we described the essence of this process in the previous paragraph), we get 2 1.17 ≈ 2.250116. In this way, 2 1,174367... ≈2 1,17 ≈2,250116 . If we take a more accurate decimal approximation of an irrational exponent, for example, , then we get a more accurate value of the original degree: 2 1,174367... ≈2 1,1743 ≈2,256833 .

Bibliography.

• Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics Zh textbook for 5 cells. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: a textbook for 7 cells. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8 cells. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: a textbook for 9 cells. educational institutions.
• Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).

Power formulas used in the process of reducing and simplifying complex expressions, in solving equations and inequalities.

Number c is n-th power of a number a when:

Operations with degrees.

1. Multiplying degrees with the same base, their indicators add up:

a ma n = a m + n .

2. In the division of degrees with the same base, their indicators are subtracted:

3. The degree of the product of 2 or more factors is equal to the product of the degrees of these factors:

(abc…) n = a n b n c n …

4. The degree of a fraction is equal to the ratio of the degrees of the dividend and the divisor:

(a/b) n = a n / b n .

5. Raising a power to a power, the exponents are multiplied:

(am) n = a m n .

Each formula above is correct in the directions from left to right and vice versa.

For example. (2 3 5/15)² = 2² 3² 5²/15² = 900/225 = 4.

Operations with roots.

1. The root of the product of several factors is equal to the product of the roots of these factors:

2. The root of the ratio is equal to the ratio of the dividend and the divisor of the roots:

3. When raising a root to a power, it is enough to raise the root number to this power:

4. If we increase the degree of the root in n once and at the same time raise to n th power is a root number, then the value of the root will not change:

5. If we decrease the degree of the root in n root at the same time n th degree from the radical number, then the value of the root will not change:

Degree with a negative exponent. The degree of a number with a non-positive (integer) exponent is defined as one divided by the degree of the same number with an exponent equal to the absolute value of the non-positive exponent:

Formula a m:a n = a m - n can be used not only for m> n, but also at m< n.

For example. a4:a 7 = a 4 - 7 = a -3.

To formula a m:a n = a m - n became fair at m=n, you need the presence of the zero degree.

Degree with zero exponent. The power of any non-zero number with a zero exponent is equal to one.

For example. 2 0 = 1,(-5) 0 = 1,(-3/5) 0 = 1.

Degree with a fractional exponent. To raise a real number a to a degree m/n, you need to extract the root n th degree of m th power of this number a.