One of the main characteristics in algebra, and in all mathematics, is degree. Of course, in the 21st century, all calculations can be done on an online calculator, but it is better for brain development to learn how to do it yourself.

In this article we will look at the most important questions related to this definition. Namely, let’s understand what it is in general and what its main functions are, what properties there are in mathematics.

Let's look at examples of what the calculation looks like and what the basic formulas are. Let's look at the main types of quantities and how they differ from other functions.

Let's understand how to solve using this quantity various tasks. We will show with examples how to raise to the zero power, irrational, negative, etc.

Online exponentiation calculator

What is a power of a number

What is meant by the expression “raise a number to a power”?

The power n of a number is the product of factors of magnitude a n times in a row.

Mathematically it looks like this:

a n = a * a * a * …a n .

For example:

• 2 3 = 2 in the third degree. = 2 * 2 * 2 = 8;
• 4 2 = 4 to step. two = 4 * 4 = 16;
• 5 4 = 5 to step. four = 5 * 5 * 5 * 5 = 625;
• 10 5 = 10 in 5 steps. = 10 * 10 * 10 * 10 * 10 = 100000;
• 10 4 = 10 in 4 steps. = 10 * 10 * 10 * 10 = 10000.

Below is a table of squares and cubes from 1 to 10.

Table of degrees from 1 to 10

Below are the results of construction natural numbers to positive powers – “from 1 to 100”.

Ch-lo	2nd st.	3rd stage
1	1	1
2	4	8
3	9	27
4	16	64
5	25	125
6	36	216
7	49	343
8	64	512
9	81	279
10	100	1000

Properties of degrees

What is characteristic of such a mathematical function? Let's look at the basic properties.

Scientists have established the following signs characteristic of all degrees:

• a n * a m = (a) (n+m) ;
• a n: a m = (a) (n-m) ;
• (a b) m =(a) (b*m) .

Let's check with examples:

2 3 * 2 2 = 8 * 4 = 32. On the other hand, 2 5 = 2 * 2 * 2 * 2 * 2 =32.

Similarly: 2 3: 2 2 = 8 / 4 =2. Otherwise 2 3-2 = 2 1 =2.

(2 3) 2 = 8 2 = 64. What if it’s different? 2 6 = 2 * 2 * 2 * 2 * 2 * 2 = 32 * 2 = 64.

As you can see, the rules work.

But what about with addition and subtraction? It's simple. Exponentiation is performed first, and then addition and subtraction.

Let's look at examples:

• 3 3 + 2 4 = 27 + 16 = 43;
• 5 2 – 3 2 = 25 – 9 = 16. Please note: the rule will not hold if you subtract first: (5 – 3) 2 = 2 2 = 4.

But in this case, you need to calculate the addition first, since there are actions in parentheses: (5 + 3) 3 = 8 3 = 512.

How to produce calculations in more complex cases? The order is the same:

• if there are brackets, you need to start with them;
• then exponentiation;
• then perform the operations of multiplication and division;
• after addition, subtraction.

There are specific properties that are not characteristic of all degrees:

1. The nth root of the number a to the degree m will be written as: a m / n.
2. When raising a fraction to a power: both the numerator and its denominator are subject to this procedure.
3. When constructing a work different numbers to a power, the expression will correspond to the product of these numbers to the given power. That is: (a * b) n = a n * b n .
4. When raising a number to a negative power, you need to divide 1 by a number in the same century, but with a “+” sign.
5. If the denominator of a fraction is to a negative power, then this expression will be equal to the product of the numerator and the denominator to a positive power.
6. Any number to the power 0 = 1, and to the power. 1 = to yourself.

These rules are important in some cases; we will consider them in more detail below.

Degree with a negative exponent

What to do with a minus degree, i.e. when the indicator is negative?

Based on properties 4 and 5(see point above), it turns out:

A (- n) = 1 / A n, 5 (-2) = 1 / 5 2 = 1 / 25.

And vice versa:

1 / A (- n) = A n, 1 / 2 (-3) = 2 3 = 8.

What if it's a fraction?

(A / B) (- n) = (B / A) n, (3 / 5) (-2) = (5 / 3) 2 = 25 / 9.

Degree with natural indicator

It is understood as a degree with exponents equal to integers.

Things to remember:

A 0 = 1, 1 0 = 1; 2 0 = 1; 3.15 0 = 1; (-4) 0 = 1...etc.

A 1 = A, 1 1 = 1; 2 1 = 2; 3 1 = 3...etc.

In addition, if (-a) 2 n +2 , n=0, 1, 2...then the result will be with a “+” sign. If a negative number raised to an odd power, then vice versa.

General properties and that's it specific signs, described above, are also characteristic of them.

Fractional degree

This type can be written as a scheme: A m / n. Read as: the nth root of the number A to the power m.

You can do whatever you want with a fractional indicator: reduce it, split it into parts, raise it to another power, etc.

Degree with irrational exponent

Let α be an irrational number and A ˃ 0.

To understand the essence of a degree with such an indicator, Let's look at different possible cases:

• A = 1. The result will be equal to 1. Since there is an axiom - 1 in all powers is equal to one;

А r 1 ˂ А α ˂ А r 2 , r 1 ˂ r 2 – rational numbers;

• 0˂А˂1.

In this case, it’s the other way around: A r 2 ˂ A α ˂ A r 1 under the same conditions as in the second paragraph.

For example, the exponent is the number π. It's rational.

r 1 – in this case equals 3;

r 2 – will be equal to 4.

Then, for A = 1, 1 π = 1.

A = 2, then 2 3 ˂ 2 π ˂ 2 4, 8 ˂ 2 π ˂ 16.

A = 1/2, then (½) 4 ˂ (½) π ˂ (½) 3, 1/16 ˂ (½) π ˂ 1/8.

Such degrees are characterized by all the mathematical operations and specific properties described above.

Conclusion

Let's summarize - what are these quantities needed for, what are the advantages of such functions? Of course, first of all, they simplify the life of mathematicians and programmers when solving examples, since they allow them to minimize calculations, shorten algorithms, systematize data, and much more.

Where else can this knowledge be useful? In any working specialty: medicine, pharmacology, dentistry, construction, technology, engineering, design, etc.

Expressions, expression conversion

Power expressions (expressions with powers) and their transformation

In this article we will talk about converting expressions with powers. First, we will focus on transformations that are performed with expressions of any kind, including power expressions, such as opening parentheses and bringing similar terms. And then we will analyze the transformations inherent specifically in expressions with degrees: working with the base and exponent, using the properties of degrees, etc.

Page navigation.

What are power expressions?

The term “power expressions” practically does not appear in school mathematics textbooks, but it appears quite often in collections of problems, especially those intended for preparation for the Unified State Exam and the Unified State Exam, for example. After analyzing the tasks in which it is necessary to perform any actions with power expressions, it becomes clear that power expressions are understood as expressions containing powers in their entries. Therefore, you can accept the following definition for yourself:

Definition.

Power expressions are expressions containing degrees.

Let's give examples of power expressions. Moreover, we will present them according to how the development of views on from degree to degree occurs. natural indicator to a degree with a real exponent.

As is known, first one gets acquainted with the power of a number with a natural exponent; at this stage, the first simplest power expressions of the type 3 2, 7 5 +1, (2+1) 5, (−0.1) 4, 3 a 2 appear −a+a 2 , x 3−1 , (a 2) 3 etc.

A little later, the power of a number with an integer exponent is studied, which leads to the appearance of power expressions with negative integer powers, like the following: 3 −2, , a −2 +2 b −3 +c 2 .

In high school they return to degrees. There a degree with a rational exponent is introduced, which entails the appearance of the corresponding power expressions: , , and so on. Finally, degrees with irrational exponents and expressions containing them are considered: , .

The matter is not limited to the listed power expressions: further the variable penetrates into the exponent, and, for example, the following expressions arise: 2 x 2 +1 or . And after getting acquainted with , expressions with powers and logarithms begin to appear, for example, x 2·lgx −5·x lgx.

So, we have dealt with the question of what power expressions represent. Next we will learn to convert them.

Main types of transformations of power expressions

With power expressions, you can perform any of the basic identity transformations of expressions. For example, you can expand the brackets, replace numeric expressions their values, give similar terms, etc. Naturally, in this case, it is necessary to follow the accepted procedure for performing actions. Let's give examples.

Example.

Calculate the value of the power expression 2 3 ·(4 2 −12) .

Solution.

According to the order of execution of actions, first perform the actions in brackets. There, firstly, we replace the power 4 2 with its value 16 (if necessary, see), and secondly, we calculate the difference 16−12=4. We have 2 3 ·(4 2 −12)=2 3 ·(16−12)=2 3 ·4.

In the resulting expression, we replace the power 2 3 with its value 8, after which we calculate the product 8·4=32. This is the desired value.

So, 2 3 ·(4 2 −12)=2 3 ·(16−12)=2 3 ·4=8·4=32.

Answer:

2 3 ·(4 2 −12)=32.

Example.

Simplify expressions with powers 3 a 4 b −7 −1+2 a 4 b −7.

Solution.

Obviously, this expression contains similar terms 3·a 4 ·b −7 and 2·a 4 ·b −7 , and we can present them: .

Answer:

3 a 4 b −7 −1+2 a 4 b −7 =5 a 4 b −7 −1.

Example.

Express an expression with powers as a product.

Solution.

You can cope with the task by representing the number 9 as a power of 3 2 and then using the formula for abbreviated multiplication - difference of squares:

Answer:

There are also a number of identical transformations inherent specifically in power expressions. We will analyze them further.

Working with base and exponent

There are degrees whose base and/or exponent are not just numbers or variables, but some expressions. As an example, we give the entries (2+0.3·7) 5−3.7 and (a·(a+1)−a 2) 2·(x+1) .

When working with such expressions, you can replace both the expression in the base of the degree and the expression in the exponent with an identically equal expression in the ODZ of its variables. In other words, according to the rules known to us, we can separately transform the base of the degree and separately the exponent. It is clear that as a result of this transformation, an expression will be obtained that is identically equal to the original one.

Such transformations allow us to simplify expressions with powers or achieve other goals we need. For example, in the power expression mentioned above (2+0.3 7) 5−3.7, you can perform operations with the numbers in the base and exponent, which will allow you to move to the power 4.1 1.3. And after opening the brackets and bringing similar terms to the base of the degree (a·(a+1)−a 2) 2·(x+1) we obtain a power expression more simple type a 2·(x+1) .

Using Degree Properties

One of the main tools for transforming expressions with powers is equalities that reflect . Let us recall the main ones. For any positive numbers a and b and arbitrary real numbers r and s, the following properties of powers are true:

• a r ·a s =a r+s ;
• a r:a s =a r−s ;
• (a·b) r =a r ·b r ;
• (a:b) r =a r:b r ;
• (a r) s =a r·s .

Note that for natural, integer, and positive exponents, the restrictions on the numbers a and b may not be so strict. For example, for natural numbers m and n the equality a m ·a n =a m+n is true not only for positive a, but also for negative a, and for a=0.

At school, the main focus when transforming power expressions is on the ability to choose the appropriate property and apply it correctly. In this case, the bases of degrees are usually positive, which allows the properties of degrees to be used without restrictions. The same applies to the transformation of expressions containing variables in the bases of powers - the range of permissible values ​​of variables is usually such that the bases take only positive values ​​on it, which allows you to freely use the properties of powers. In general, you need to constantly ask yourself whether it is possible to use any property of degrees in this case, because inaccurate use of properties can lead to a narrowing of the educational value and other troubles. These points are discussed in detail and with examples in the article transformation of expressions using properties of degrees. Here we will limit ourselves to considering a few simple examples.

Example.

Express the expression a 2.5 ·(a 2) −3:a −5.5 as a power with base a.

Solution.

First, we transform the second factor (a 2) −3 using the property of raising a power to a power: (a 2) −3 =a 2·(−3) =a −6. The original power expression will take the form a 2.5 ·a −6:a −5.5. Obviously, it remains to use the properties of multiplication and division of powers with the same base, we have
a 2.5 ·a −6:a −5.5 =
a 2.5−6:a −5.5 =a −3.5:a −5.5 =
a −3.5−(−5.5) =a 2 .

Answer:

a 2.5 ·(a 2) −3:a −5.5 =a 2.

Properties of powers when transforming power expressions are used both from left to right and from right to left.

Example.

Find the value of the power expression.

Solution.

The equality (a·b) r =a r ·b r, applied from right to left, allows us to move from the original expression to a product of the form and further. And when multiplying powers with on the same grounds the indicators add up: .

It was possible to transform the original expression in another way:

Answer:

.

Example.

Given the power expression a 1.5 −a 0.5 −6, introduce a new variable t=a 0.5.

Solution.

The degree a 1.5 can be represented as a 0.5 3 and then, based on the property of the degree to the degree (a r) s =a r s, applied from right to left, transform it to the form (a 0.5) 3. Thus, a 1.5 −a 0.5 −6=(a 0.5) 3 −a 0.5 −6. Now it’s easy to introduce a new variable t=a 0.5, we get t 3 −t−6.

Answer:

t 3 −t−6 .

Converting fractions containing powers

Power expressions can contain or represent fractions with powers. Any of the basic transformations of fractions that are inherent in fractions of any kind are fully applicable to such fractions. That is, fractions that contain powers can be reduced, reduced to a new denominator, worked separately with their numerator and separately with the denominator, etc. To illustrate these words, consider solutions to several examples.

Example.

Simplify power expression .

Solution.

This power expression is a fraction. Let's work with its numerator and denominator. In the numerator we open the brackets and simplify the resulting expression using the properties of powers, and in the denominator we present similar terms:

And let’s also change the sign of the denominator by placing a minus in front of the fraction: .

Answer:

.

Reducing fractions containing powers to a new denominator is carried out similarly to reducing rational fractions to a new denominator. In this case, an additional factor is also found and the numerator and denominator of the fraction are multiplied by it. When performing this action, it is worth remembering that reduction to a new denominator can lead to a narrowing of the VA. To prevent this from happening, it is necessary that the additional factor does not go to zero for any values ​​of the variables from the ODZ variables for the original expression.

Example.

Reduce the fractions to a new denominator: a) to denominator a, b) to the denominator.

Solution.

a) In this case, it is quite easy to figure out what additional multiplier helps to achieve desired result. This is a multiplier of a 0.3, since a 0.7 ·a 0.3 =a 0.7+0.3 =a. Note that in the range of permissible values ​​of the variable a (this is the set of all positive real numbers), the power of a 0.3 does not vanish, therefore, we have the right to multiply the numerator and denominator of a given fraction by this additional factor:

b) Taking a closer look at the denominator, you will find that

and multiplying this expression by will give the sum of cubes and , that is, . And this is the new denominator to which we need to reduce the original fraction.

This is how we found an additional factor. In the range of acceptable values ​​of the variables x and y, the expression does not vanish, therefore, we can multiply the numerator and denominator of the fraction by it:

Answer:

A) , b) .

There is also nothing new in reducing fractions containing powers: the numerator and denominator are represented as a number of factors, and the same factors of the numerator and denominator are reduced.

Example.

Reduce the fraction: a) , b) .

Solution.

a) Firstly, the numerator and denominator can be reduced by the numbers 30 and 45, which is equal to 15. It is also obviously possible to perform a reduction by x 0.5 +1 and by . Here's what we have:

b) In this case, identical factors in the numerator and denominator are not immediately visible. To obtain them, you will have to perform preliminary transformations. In this case, they consist in factoring the denominator using the difference of squares formula:

Answer:

A)

b) .

Converting fractions to a new denominator and reducing fractions are mainly used to do things with fractions. Actions are performed according to known rules. When adding (subtracting) fractions, they are reduced to a common denominator, after which the numerators are added (subtracted), but the denominator remains the same. The result is a fraction whose numerator is the product of the numerators, and the denominator is the product of the denominators. Division by a fraction is multiplication by its inverse.

Example.

Follow the steps .

Solution.

First, we subtract the fractions in parentheses. To do this, we bring them to a common denominator, which is , after which we subtract the numerators:

Now we multiply the fractions:

Obviously, it is possible to reduce by a power of x 1/2, after which we have .

You can also simplify the power expression in the denominator by using the difference of squares formula: .

Answer:

Example.

Simplify the Power Expression .

Solution.

Obviously, this fraction can be reduced by (x 2.7 +1) 2, this gives the fraction . It is clear that something else needs to be done with the powers of X. To do this, we transform the resulting fraction into a product. This gives us the opportunity to take advantage of the property of dividing powers with the same bases: . And at the end of the process we move from the last product to the fraction.

Answer:

.

And let us also add that it is possible, and in many cases desirable, to transfer factors with negative exponents from the numerator to the denominator or from the denominator to the numerator, changing the sign of the exponent. Such transformations often simplify further actions. For example, a power expression can be replaced by .

Converting expressions with roots and powers

Often, in expressions in which some transformations are required, roots with fractional exponents are also present along with powers. To convert such an expression to the right type, in most cases it is enough to go only to roots or only to powers. But since it is more convenient to work with powers, they usually move from roots to powers. However, it is advisable to carry out such a transition when the ODZ of variables for the original expression allows you to replace the roots with powers without the need to refer to the module or split the ODZ into several intervals (we discussed this in detail in the article transition from roots to powers and back After getting acquainted with the degree with a rational exponent a degree with an irrational exponent is introduced, which allows us to talk about a degree with an arbitrary real exponent. At this stage, the school begins to study exponential function, which is analytically given by a power, the base of which is a number, and the exponent is a variable. So we are faced with power expressions containing numbers in the base of the power, and in the exponent - expressions with variables, and naturally the need arises to perform transformations of such expressions.

It should be said that the transformation of expressions of the indicated type usually has to be performed when solving exponential equations And exponential inequalities , and these conversions are quite simple. In the overwhelming majority of cases, they are based on the properties of the degree and are aimed, for the most part, at introducing a new variable in the future. The equation will allow us to demonstrate them 5 2 x+1 −3 5 x 7 x −14 7 2 x−1 =0.

Firstly, powers, in the exponents of which is the sum of a certain variable (or expression with variables) and a number, are replaced by products. This applies to the first and last terms of the expression on the left side:
5 2 x 5 1 −3 5 x 7 x −14 7 2 x 7 −1 =0,
5 5 2 x −3 5 x 7 x −2 7 2 x =0.

Next, both sides of the equality are divided by the expression 7 2 x, which on the ODZ of the variable x for the original equation takes only positive values ​​(this is a standard technique for solving equations of this type, we are not talking about it now, so focus on subsequent transformations of expressions with powers ):

Now we can cancel fractions with powers, which gives .

Finally, the ratio of powers with the same exponents is replaced by powers of relations, resulting in the equation , which is equivalent . The transformations made allow us to introduce a new variable, which reduces the solution of the original exponential equation to the solution of a quadratic equation

I. V. Boykov, L. D. Romanova Collection of tasks for preparing for the Unified State Exam. Part 1. Penza 2003.

Lesson type: lesson of generalization and systematization of knowledge

Goals:

• educational– repeat the definition of a degree, the rules for multiplying and dividing degrees, raising a degree to a power, consolidate the skills of solving examples containing degrees,
• developing– development of students’ logical thinking, interest in the material being studied,
• raising– fostering a responsible attitude to learning, a culture of communication, and a sense of collectivism.

Equipment: computer, multimedia projector, interactive whiteboard, presentation of “Degrees” for mental calculation, cards with tasks, handouts.

Lesson plan:

1. Organizing time.
2. Repetition of rules
3. Verbal counting.
4. Historical reference.
5. Work at the board.
6. Physical education minute.
7. Working on an interactive whiteboard.
8. Independent work.
9. Homework.
10. Summing up the lesson.

During the classes

I. Organizational moment

Communicate the topic and objectives of the lesson.

In previous lessons you discovered amazing world degrees, learned to multiply and divide degrees, and raise them to a power. Today we must consolidate the acquired knowledge by solving examples.

II. Repetition of rules(orally)

1. Give the definition of degree with a natural exponent? (Power of number A with a natural exponent greater than 1 is called a product n factors, each of which is equal A.)
2. How to multiply two powers? (To multiply powers with the same bases, you must leave the base the same and add the exponents.)
3. How to divide degree by degree? (To divide powers with the same bases, you need to leave the base the same and subtract the exponents.)
4. How to raise a product to a power? (To raise a product to a power, you need to raise each factor to that power)
5. How to raise a degree to a power? (To raise a power to a power, you need to leave the base the same and multiply the exponents)

III. Verbal counting(by multimedia)

IV. Historical reference

All problems are from the Ahmes papyrus, which was written around 1650 BC. e. related to construction practice, demarcation of land plots, etc. Tasks are grouped by topic. These are mainly tasks on finding the areas of a triangle, quadrilaterals and a circle, various operations with integers and fractions, proportional division, finding ratios, and there is also raising in different degrees, solving equations of the first and second degree with one unknown.

There is a complete lack of any explanation or evidence. The desired result is either given directly or a short algorithm for calculating it is given. This method of presentation, typical of science in the countries of the ancient East, suggests that mathematics there developed through generalizations and guesses that did not form any general theory. However, the papyrus contains a number of evidence that Egyptian mathematicians knew how to extract roots and raise to powers, solve equations, and even mastered the rudiments of algebra.

V. Work at the board

Find the meaning of the expression in a rational way:

Calculate the value of the expression:

VI. Physical education minute

1. for eyes
2. for the neck
3. for hands
4. for the torso
5. for legs

VII. Problem solving(with display on the interactive whiteboard)

Is the root of the equation a positive number?

a) 3x + (-0.1) 7 = (-0.496) 4 (x > 0)

b) (10.381) 5 = (-0.012) 3 - 2x (x< 0)

VIII. Independent work

IX. Homework

X. Summing up the lesson

Analysis of results, announcement of grades.

We will use the acquired knowledge about degrees when solving equations and problems in high school; they are also often found in the Unified State Exam.

I. Work n factors, each of which is equal A called n-th power of the number A and is designated An.

Examples. Write the product as a degree.

1) mmmm; 2) aaabb; 3) 5 5 5 5 ccc; 4) ppkk+pppk-ppkkk.

Solution.

1) mmmm=m 4, since, by definition of a degree, the product of four factors, each of which is equal m, will fourth power of m.

2) aaabb=a 3 b 2 ; 3) 5·5·5·5·ccc=5 4 c 3 ; 4) ppkk+pppk-ppkkk=p 2 k 2 +p 3 k-p 2 k 3.

II. The action by which the product of several equal factors is found is called exponentiation. The number that is raised to a power is called the base of the power. The number that shows to what power the base is raised is called the exponent. So, An- degree, A– the basis of the degree, n– exponent. For example:

2 3 — it's a degree. Number 2 is the base of the degree, the exponent is equal to 3 . Degree value 2 3 equals 8, because 2 3 =2·2·2=8.

Examples. Write the following expressions without the exponent.

5) 4 3; 6) a 3 b 2 c 3 ; 7) a 3 -b 3 ; 8) 2a 4 +3b 2 .

Solution.

5) 4 3 = 4·4·4 ; 6) a 3 b 2 c 3 = aaabbccc; 7) a 3 -b 3 = aaa-bbb; 8) 2a 4 +3b 2 = 2aaaa+3bb.

III. and 0 =1 Any number (except zero) to the zero power is equal to one. For example, 25 0 =1.
IV. a 1 =aAny number to the first power is equal to itself.

V. a m∙ a n= a m + n When multiplying powers with the same bases, the base is left the same, and the exponents folded

Examples. Simplify:

9) a·a 3 ·a 7 ; 10) b 0 +b 2 b 3 ; 11) c 2 ·c 0 ·c·c 4 .

Solution.

9) a·a 3 ·a 7=a 1+3+7 =a 11 ; 10) b 0 +b 2 b 3 = 1+b 2+3 =1+b 5 ;

11) c 2 c 0 c c 4 = 1 c 2 c c 4 =c 2+1+4 =c 7 .

VI. a m: a n= a m - nWhen dividing powers with the same base, the base is left the same, and the exponent of the divisor is subtracted from the exponent of the dividend.

Examples. Simplify:

12) a 8:a 3 ; 13) m 11:m 4 ; 14) 5 6:5 4 .

12)a 8:a 3=a 8-3 =a 5 ; 13)m 11:m 4=m 11-4 =m 7; 14 ) 5 6:5 4 =5 2 =5·5=25.

VII. (a m) n= a mn When raising a power to a power, the base is left the same, and the exponents are multiplied.

Examples. Simplify:

15) (a 3) 4 ; 16) (c 5) 2.

15) (a 3) 4=a 3·4 =a 12 ; 16) (c 5) 2=c 5 2 =c 10.

note, which, since the product does not change from rearranging the factors, That:

15) (a 3) 4 = (a 4) 3 ; 16) (c 5) 2 = (c 2) 5 .

VI II. (a∙b) n =a n ∙b n When raising a product to a power, each of the factors is raised to that power.

Examples. Simplify:

17) (2a 2) 5 ; 18) 0.2 6 5 6 ; 19) 0.25 2 40 2.

Solution.

17) (2a 2) 5=2 5 ·a 2·5 =32a 10 ; 18) 0.2 6 5 6=(0.2·5) 6 =1 6 =1;

19) 0.25 2 40 2=(0.25·40) 2 =10 2 =100.

IX. When raising a fraction to a power, both the numerator and denominator of the fraction are raised to that power.

Examples. Simplify:

Solution.

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Lesson on the topic: "Rules of multiplication and division of powers with the same and different exponents. Examples"

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Manual for the textbook Yu.N. Makarycheva Manual for the textbook by A.G. Mordkovich

Purpose of the lesson: learn to perform operations with powers of numbers.

First, let's remember the concept of "power of number". An expression of the form $\underbrace( a * a * \ldots * a )_(n)$ can be represented as $a^n$.

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

This equality is called “recording the degree as a product.” It will help us determine how to multiply and divide powers.
Remember:
a– the basis of the degree.
n– exponent.
If n=1, which means the number A took once and accordingly: $a^n= 1$.
If n= 0, then $a^0= 1$.

We can find out why this happens 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 get $a^n * a^m$, we write the degrees as a product: $\underbrace( a * a * \ldots * a )_(n) * \underbrace( a * a * \ldots * a )_(m )$.
The figure shows that the number A have taken n+m times, then $a^n * a^m = a^(n + m)$.

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

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

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

So $a^n * b^n= (a * b)^n$.

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

Division rules

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

So, we need $\frac(a^n)(a^m)$, Where n>m.

Let's write the degrees as a fraction:

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

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

Now let's reduce the fraction.

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

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

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

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

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

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

For convenience, let's imagine.

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

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