Raising the difference to a power. Details about degrees and exponentiation

Exponentiation is an operation closely related to multiplication; this operation is the result of repeatedly multiplying a number by itself. Let's represent it with 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 main ones: Addition, Subtraction, Multiplication, Division.

Raising a number to a power

Raising a number to a power is not a complicated operation. It is related to multiplication in a similar way to the relationship between multiplication and addition. The notation an is a short notation of the nth number of numbers “a” multiplied by each other.

Consider exponentiation using 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 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) is equal to one hundred 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 given below. There is nothing extra natural in this, the main thing is to understand the essence and then they will not only be remembered, but will also seem easy.

Raising a monomial to a power

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

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

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.

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

Raising to a negative power

A negative power is the reciprocal of a number. What is the reciprocal number? The reciprocal of any number X 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. Simple isn't it?

Raising to a fractional power

Let's start by looking at the issue with a specific example. 43/2. What does degree 3/2 mean? 3 – numerator, means raising a number (in this case 4) to a cube. The number 2 is the denominator; it is the extraction of the second root of a number (in this case, 4).

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

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

Raising a root to a power

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

If (√x)^4. Then (√x)^4=x^2. To check the solution, we convert the expression into an expression with a fractional power. 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 into an expression with a fractional power. If the fraction does not cancel, then this is the answer, provided that the root of the given number is not isolated.

Raising a complex number to the power

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 a number that, when squared, gives the number -1.

Let's look at 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

Using our calculator, you can calculate the raising of a number to a power:

Exponentiation 7th grade

Schoolchildren begin raising to a power only in the seventh grade.

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

Examples for solution:

Exponentiation presentation

Presentation on raising to powers designed for seventh graders. The presentation may clarify some unclear points, but these points will probably not be cleared up thanks to our article.

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When the number multiplies itself to myself, work called degree.

So 2.2 = 4, square or second power of 2
2.2.2 = 8, cube or third power.
2.2.2.2 = 16, fourth degree.

Also, 10.10 = 100, the second power of 10.
10.10.10 = 1000, third degree.
10.10.10.10 = 10000 fourth power.

And a.a = aa, second power of a
a.a.a = aaa, third power of a
a.a.a.a = aaaa, fourth power of a

The original number is called root powers of this number because it is the number from which the powers were created.

However, it is not entirely convenient, especially in the case of high powers, to write down all the factors that make up the powers. Therefore, a shorthand notation method is used. The root of the degree is written only once, and on the right and a little higher near it, but in a slightly smaller font, it is written how many times the root acts as a factor. This number or letter is called exponent or degree numbers. So, a 2 is equal to a.a or aa, because the root a must be multiplied by itself twice to get the power aa. Also, a 3 means aaa, that is, here a is repeated three times as a multiplier.

The exponent of the first degree is 1, but it is not usually written down. So, a 1 is written as a.

You should not confuse degrees with coefficients. The coefficient shows how often the value is taken as Part the whole. The power shows how often a quantity is taken as factor in the work.
So, 4a = a + a + a + a. But a 4 = a.a.a.a

The power notation scheme has the peculiar advantage of allowing us to express unknown degree. For this purpose, the exponent is written instead of a number letter. In the process of solving a problem, we can obtain a quantity that we know is some degree of another magnitude. But so far we do not know whether it is a square, a cube or another, higher degree. So, in the expression a x, the exponent means that this expression has some degree, although unspecified what degree. So, b m and d n are raised to the powers of m and n. When the exponent is found, number is substituted instead of a letter. So, if m=3, then b m = b 3 ; but if m = 5, then b m =b 5.

The method of writing values using powers is also a big advantage when using expressions. Thus, (a + b + d) 3 is (a + b + d).(a + b + d).(a + b + d), that is, the cube of the trinomial (a + b + d). But if we write this expression after raising it to a cube, it will look like
a 3 + 3a 2 b + 3a 2 d + 3ab 2 + 6abd + 3ad 2 + b 3 + d 3 .

If we take a series of powers whose exponents increase or decrease by 1, we find that the product increases by common multiplier or decreases by common divisor, and this factor or divisor is the original number that is raised to a power.

So, in the series aaaaa, aaaa, aaa, aa, a;
or a 5, a 4, a 3, a 2, a 1;
the indicators, if counted from right to left, are 1, 2, 3, 4, 5; and the difference between their values is 1. If we start on right multiply by a, we will successfully get multiple values.

So a.a = a 2 , second term. And a 3 .a = a 4
a 2 .a = a 3 , third term. a 4 .a = a 5 .

If we start left divide to a,
we get a 5:a = a 4 and a 3:a = a 2 .
a 4:a = a 3 a 2:a = a 1

But this division process can be continued further, and we get a new set of values.

So, a:a = a/a = 1. (1/a):a = 1/aa
1:a = 1/a (1/aa):a = 1/aaa.

The complete row would be: aaaaa, aaaa, aaa, aa, a, 1, 1/a, 1/aa, 1/aaa.

Or a 5, a 4, a 3, a 2, a, 1, 1/a, 1/a 2, 1/a 3.

Here are the values on right from one there is reverse values to the left of one. Therefore these degrees can be called inverse powers a. We can also say that the powers on the left are the inverses of the powers on the right.

So, 1:(1/a) = 1.(a/1) = a. And 1:(1/a 3) = a 3.

The same recording plan can be applied to polynomials. So, for a + b, we get the set,
(a + b) 3 , (a + b) 2 , (a + b), 1, 1/(a + b), 1/(a + b) 2 , 1/(a + b) 3 .

For convenience, another form of writing reciprocal powers is used.

According to this form, 1/a or 1/a 1 = a -1 . And 1/aaa or 1/a 3 = a -3 .
1/aa or 1/a 2 = a -2 . 1/aaaa or 1/a 4 = a -4 .

And in order to make a complete series with 1 as a total difference with exponents, a/a or 1 is considered as something that does not have a degree and is written as a 0 .

Then, taking into account the direct and inverse powers
instead of aaaa, aaa, aa, a, a/a, 1/a, 1/aa, 1/aaa, 1/aaaa
you can write a 4, a 3, a 2, a 1, a 0, a -1, a -2, a -3, a -4.
Or a +4, a +3, a +2, a +1, a 0, a -1, a -2, a -3, a -4.

And a series of only individual degrees will look like:
+4,+3,+2,+1,0,-1,-2,-3,-4.

The root of a degree can be expressed by more than one letter.

Thus, aa.aa or (aa) 2 is the second power of aa.
And aa.aa.aa or (aa) 3 is the third power of aa.

All powers of the number 1 are the same: 1.1 or 1.1.1. will be equal to 1.

Exponentiation is finding the value of any number by multiplying that number by itself. Rule for exponentiation:

Multiply the quantity by itself as many times as indicated in the power of the number.

This rule is common to all examples that may arise during the process of exponentiation. But it is right to give an explanation of how it applies to particular cases.

If only one term is raised to a power, then it is multiplied by itself as many times as indicated by the exponent.

The fourth power of a is a 4 or aaaa. (Art. 195.)
The sixth power of y is y 6 or yyyyyy.
The Nth power of x is x n or xxx..... n times repeated.

If it is necessary to raise an expression of several terms to a power, the principle that the power of the product of several factors is equal to the product of these factors raised to a power.

So (ay) 2 =a 2 y 2 ; (ay) 2 = ay.ay.
But ay.ay = ayay = aayy = a 2 y 2 .
So, (bmx) 3 = bmx.bmx.bmx = bbbmmmxxx = b 3 m 3 x 3 .

Therefore, in finding the power of a product, we can either operate with the entire product at once, or we can operate with each factor separately, and then multiply their values with the powers.

Example 1. The fourth power of dhy is (dhy) 4, or d 4 h 4 y 4.

Example 2. The third power is 4b, there is (4b) 3, or 4 3 b 3, or 64b 3.

Example 3. The Nth power of 6ad is (6ad) n or 6 n a n d n.

Example 4. The third power of 3m.2y is (3m.2y) 3, or 27m 3 .8y 3.

The degree of a binomial, consisting of terms connected by + and -, is calculated by multiplying its terms. Yes,

(a + b) 1 = a + b, first degree.
(a + b) 1 = a 2 + 2ab + b 2, second power (a + b).
(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3, third power.
(a + b) 4 = a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4, fourth power.

The square of a - b is a 2 - 2ab + b 2.

The square of a + b + h is a 2 + 2ab + 2ah + b 2 + 2bh + h 2

Exercise 1. Find the cube a + 2d + 3

Exercise 2. Find the fourth power of b + 2.

Exercise 3. Find the fifth power of x + 1.

Exercise 4. Find the sixth power 1 - b.

Sum squares amounts And differences binomials occur so often in algebra that it is necessary to know them very well.

If we multiply a + h by itself or a - h by itself,
we get: (a + h)(a + h) = a 2 + 2ah + h 2 also, (a - h)(a - h) = a 2 - 2ah + h 2 .

This shows that in each case, the first and last terms are the squares of a and h, and the middle term is twice the product of a and h. From here, the square of the sum and difference of binomials can be found using the following rule.

The square of a binomial, both terms of which are positive, is equal to the square of the first term + twice the product of both terms + the square of the last term.

Square differences binomials is equal to the square of the first term minus twice the product of both terms plus the square of the second term.

Example 1. Square 2a + b, there is 4a 2 + 4ab + b 2.

Example 2. Square ab + cd, there is a 2 b 2 + 2abcd + c 2 d 2.

Example 3. Square 3d - h, there is 9d 2 + 6dh + h 2.

Example 4. The square a - 1 is a 2 - 2a + 1.

For a method for finding higher powers of binomials, see the following sections.

In many cases it is effective to write degrees without multiplication.

So, the square of a + b is (a + b) 2.
The Nth power of bc + 8 + x is (bc + 8 + x) n

In such cases, the parentheses cover All members under degree.

But if the root of the degree consists of several multipliers, the parentheses may cover the entire expression, or may be applied separately to the factors depending on convenience.

Thus, the square (a + b)(c + d) is either [(a + b).(c + d)] 2 or (a + b) 2 .(c + d) 2.

For the first of these expressions, the result is the square of the product of two factors, and for the second, the result is the product of their squares. But they are equal to each other.

Cube a.(b + d), is 3, or a 3.(b + d) 3.

The sign in front of the members involved must also be taken into account. It is very important to remember that when the root of a degree is positive, all its positive powers are also positive. But when the root is negative, the values with odd powers are negative, while the values even degrees are positive.

The second degree (- a) is +a 2
The third degree (-a) is -a 3
The fourth power (-a) is +a 4
The fifth power (-a) is -a 5

Hence any odd the degree has the same sign as the number. But even the degree is positive regardless of whether the number has a negative or positive sign.
So, +a.+a = +a 2
And -a.-a = +a 2

A quantity that has already been raised to a power is raised to a power again by multiplying the exponents.

The third power of a 2 is a 2.3 = a 6.

For a 2 = aa; cube aa is aa.aa.aa = aaaaaa = a 6 ; which is the sixth power of a, but the third power of a 2.

The fourth power of a 3 b 2 is a 3.4 b 2.4 = a 12 b 8

The third power of 4a 2 x is 64a 6 x 3.

The fifth power of (a + b) 2 is (a + b) 10.

The Nth power of a 3 is a 3n

The Nth power of (x - y) m is (x - y) mn

(a 3 .b 3) 2 = a 6 .b 6

(a 3 b 2 h 4) 3 = a 9 b 6 h 12

The rule applies equally to negative degrees.

Example 1. The third power of a -2 is a -3.3 =a -6.

For a -2 = 1/aa, and the third power of this
(1/aa).(1/aa).(1/aa) = 1/aaaaaa = 1/a 6 = a -6

The fourth power of a 2 b -3 is a 8 b -12 or a 8 /b 12.

The square is b 3 x -1, there is b 6 x -2.

The Nth power of ax -m is x -mn or 1/x.

However, we must remember here that if the sign previous degree is "-", then it must be changed to "+" whenever the degree is an even number.

Example 1. The square -a 3 is +a 6. The square of -a 3 is -a 3 .-a 3, which, according to the rules of signs in multiplication, is +a 6.

2. But the cube -a 3 is -a 9. For -a 3 .-a 3 .-a 3 = -a 9 .

3. The Nth power -a 3 is a 3n.

Here the result can be positive or negative depending on whether n is even or odd.

If fraction is raised to a power, then the numerator and denominator are raised to a power.

The square of a/b is a 2 /b 2 . According to the rule for multiplying fractions,
(a/b)(a/b) = aa/bb = a 2 b 2

The second, third and nth powers of 1/a are 1/a 2, 1/a 3 and 1/a n.

Examples binomials, in which one of the terms is a fraction.

1. Find the square of x + 1/2 and x - 1/2.
(x + 1/2) 2 = x 2 + 2.x.(1/2) + 1/2 2 = x 2 + x + 1/4
(x - 1/2) 2 = x 2 - 2.x.(1/2) + 1/2 2 = x 2 - x + 1/4

2. The square of a + 2/3 is a 2 + 4a/3 + 4/9.

3. Square x + b/2 = x 2 + bx + b 2 /4.

4 The square of x - b/m is x 2 - 2bx/m + b 2 /m 2 .

It was previously shown that fractional coefficient can be moved from the numerator to the denominator or from the denominator to the numerator. Using the scheme for writing reciprocal powers, it is clear that any multiplier can also be moved, if the sign of the degree is changed.

So, in the fraction ax -2 /y, we can move x from the numerator to the denominator.
Then ax -2 /y = (a/y).x -2 = (a/y).(1/x 2 = a/yx 2.

In the fraction a/by 3, we can move y from the denominator to the numerator.
Then a/by 2 = (a/b).(1/y 3) = (a/b).y -3 = ay -3 /b.

In the same way, we can move a factor that has a positive exponent to the numerator or a factor with a negative exponent to the denominator.

So, ax 3 /b = a/bx -3. For x 3 the inverse is x -3 , which is x 3 = 1/x -3 .

Therefore, the denominator of any fraction can be completely removed, or the numerator can be reduced to one, without changing the meaning of the expression.

So, a/b = 1/ba -1 , or ab -1 .

can be found using multiplication. For example: 5+5+5+5+5+5=5x6. Such an expression is said to be that the sum of equal terms is folded into a product. And vice versa, if we read this equality from right to left, we find that we have expanded the sum of equal terms. Similarly, you can collapse the product of several equal factors 5x5x5x5x5x5=5 6.

That is, instead of multiplying six identical factors 5x5x5x5x5x5, they write 5 6 and say “five to the sixth power.”

The expression 5 6 is a power of a number, where:

5 - degree base;

6 - exponent.

Actions by which the product of equal factors is reduced to a power are called raising to a power.

In general, a degree with base “a” and exponent “n” is written as follows

Raising the number a to the power n means finding the product of n factors, each of which is equal to a

If the base of the degree “a” is equal to 1, then the value of the degree for any natural number n will be equal to 1. For example, 1 5 =1, 1 256 =1

If you raise the number “a” to first degree, then we get the number a itself: a 1 = a

If you raise any number to zero degree, then as a result of calculations we get one. a 0 = 1

The second and third powers of a number are considered special. They came up with names for them: the second degree is called square the number, third - cube this number.

Any number can be raised to a power - positive, negative or zero. In this case, the following rules do not apply:

When finding the power of a positive number, the result is a positive number.

When calculating zero to the natural power, we get zero.

x m · x n = x m + n

for example: 7 1.7 7 - 0.9 = 7 1.7+(- 0.9) = 7 1.7 - 0.9 = 7 0.8

To divide powers with the same bases We do not change the base, but subtract the exponents:

x m / x n = x m - n , Where, m > n,

for example: 13 3.8 / 13 -0.2 = 13 (3.8 -0.2) = 13 3.6

When calculating raising a power to a power We do not change the base, but multiply the exponents by each other.

(at m ) n = y m n

for example: (2 3) 2 = 2 3 2 = 2 6

(X · y) n = x n · y m ,

for example:(2 3) 3 = 2 n 3 m,

When performing calculations according to raising a fraction to a power we raise the numerator and denominator of the fraction to a given power

(x/y)n = x n / y n

for example: (2 / 5) 3 = (2 / 5) · (2 / 5) · (2 / 5) = 2 3 / 5 3.

The sequence of calculations when working with expressions containing a degree.

When performing calculations of expressions without parentheses, but containing powers, first of all, they perform exponentiation, then multiplication and division, and only then addition and subtraction operations.

If you need to calculate an expression containing brackets, then first do the calculations in the brackets in the order indicated above, and then the remaining actions in the same order from left to right.

Very widely in practical calculations, ready-made tables of powers are used to simplify calculations.

We figured out what a power of a number actually is. Now we need to understand how to calculate it correctly, i.e. raise numbers to powers. In this material we will analyze the basic rules for calculating degrees in the case of integer, natural, fractional, rational and irrational exponents. All definitions will be illustrated with examples.

The concept of exponentiation

Let's start by formulating basic definitions.

Definition 1

Exponentiation is the calculation of the value of the power of a certain number.

That is, the words “calculating the value of a power” and “raising to a power” mean the same thing. So, if the problem says “Raise the number 0, 5 to the fifth power,” this should be understood as “calculate the value of the power (0, 5) 5.

Now we present the basic rules that must be followed when making such calculations.

Let's remember what a power of a number with a natural exponent is. For a power with base a and exponent n, this will be the product of the nth number of factors, each of which is equal to a. It can be written like this:

To calculate the value of a degree, you need to perform a multiplication action, that is, multiply the bases of the degree the specified number of times. The very concept of a degree with a natural exponent is based on the ability to quickly multiply. Let's give examples.

Example 1

Condition: raise - 2 to the power 4.

Solution

Using the definition above, we write: (− 2) 4 = (− 2) · (− 2) · (− 2) · (− 2) . Next, we just need to follow these steps and get 16.

Let's take a more complicated example.

Example 2

Calculate the value 3 2 7 2

Solution

This entry can be rewritten as 3 2 7 · 3 2 7 . Previously, we looked at how to correctly multiply the mixed numbers mentioned in the condition.

Let's perform these steps and get the answer: 3 2 7 · 3 2 7 = 23 7 · 23 7 = 529 49 = 10 39 49

If the problem indicates the need to raise irrational numbers to a natural power, we will need to first round their bases to the digit that will allow us to obtain an answer of the required accuracy. Let's look at an example.

Example 3

Perform the square of π.

Solution

First, let's round it to hundredths. Then π 2 ≈ (3, 14) 2 = 9, 8596. If π ≈ 3. 14159, then we get a more accurate result: π 2 ≈ (3, 14159) 2 = 9, 8695877281.

Note that the need to calculate powers of irrational numbers arises relatively rarely in practice. We can then write the answer as the power (ln 6) 3 itself, or convert if possible: 5 7 = 125 5 .

Separately, it should be indicated what the first power of a number is. Here you can simply remember that any number raised to the first power will remain itself:

This is clear from the recording .

It does not depend on the basis of the degree.

Example 4

So, (− 9) 1 = − 9, and 7 3 raised to the first power will remain equal to 7 3.

For convenience, we will examine three cases separately: if the exponent is a positive integer, if it is zero and if it is a negative integer.

In the first case, this is the same as raising to a natural power: after all, positive integers belong to the set of natural numbers. We have already talked above about how to work with such degrees.

Now let's see how to correctly raise to the zero power. For a base other than zero, this calculation always outputs 1. We previously explained that the 0th power of a can be defined for any real number not equal to 0, and a 0 = 1.

Example 5

5 0 = 1 , (- 2 , 56) 0 = 1 2 3 0 = 1

0 0 - not defined.

We are left with only the case of a degree with an integer negative exponent. We have already discussed that such degrees can be written as a fraction 1 a z, where a is any number, and z is a negative integer. We see that the denominator of this fraction is nothing more than an ordinary power with a positive integer exponent, and we have already learned how to calculate it. Let's give examples of tasks.

Example 6

Raise 2 to the power - 3.

Solution

Using the definition above, we write: 2 - 3 = 1 2 3

Let's calculate the denominator of this fraction and get 8: 2 3 = 2 · 2 · 2 = 8.

Then the answer is: 2 - 3 = 1 2 3 = 1 8

Example 7

Raise 1.43 to the -2 power.

Solution

Let's reformulate: 1, 43 - 2 = 1 (1, 43) 2

We calculate the square in the denominator: 1.43·1.43. Decimals can be multiplied in this way:

As a result, we got (1, 43) - 2 = 1 (1, 43) 2 = 1 2, 0449. All we have to do is write this result in the form of an ordinary fraction, for which we need to multiply it by 10 thousand (see the material on converting fractions).

Answer: (1, 43) - 2 = 10000 20449

A special case is raising a number to the minus first power. The value of this degree is equal to the reciprocal of the original value of the base: a - 1 = 1 a 1 = 1 a.

Example 8

Example: 3 − 1 = 1 / 3

9 13 - 1 = 13 9 6 4 - 1 = 1 6 4 .

How to raise a number to a fractional power

To perform such an operation, we need to remember the basic definition of a degree with a fractional exponent: a m n = a m n for any positive a, integer m and natural n.

Definition 2

Thus, the calculation of a fractional power must be performed in two steps: raising to an integer power and finding the root of the nth power.

We have the equality a m n = a m n , which, taking into account the properties of the roots, is usually used to solve problems in the form a m n = a n m . This means that if we raise a number a to a fractional power m / n, then first we take the nth root of a, then we raise the result to a power with an integer exponent m.

Let's illustrate with an example.

Example 9

Calculate 8 - 2 3 .

Solution

Method 1: According to the basic definition, we can represent this as: 8 - 2 3 = 8 - 2 3

Now let's calculate the degree under the root and extract the third root from the result: 8 - 2 3 = 1 64 3 = 1 3 3 64 3 = 1 3 3 4 3 3 = 1 4

Method 2. Transform the basic equality: 8 - 2 3 = 8 - 2 3 = 8 3 - 2

After this, we extract the root 8 3 - 2 = 2 3 3 - 2 = 2 - 2 and square the result: 2 - 2 = 1 2 2 = 1 4

We see that the solutions are identical. You can use it any way you like.

There are cases when the degree has an indicator expressed as a mixed number or a decimal fraction. To simplify calculations, it is better to replace it with an ordinary fraction and calculate as indicated above.

Example 10

Raise 44, 89 to the power of 2, 5.

Solution

Let's convert the value of the indicator into an ordinary fraction: 44, 89 2, 5 = 44, 89 5 2.

Now we carry out in order all the actions indicated above: 44, 89 5 2 = 44, 89 5 = 44, 89 5 = 4489 100 5 = 4489 100 5 = 67 2 10 2 5 = 67 10 5 = = 1350125107 100000 = 13 501, 25107

Answer: 13 501, 25107.

If the numerator and denominator of a fractional exponent contain large numbers, then calculating such exponents with rational exponents is a rather difficult job. It usually requires computer technology.

Let us separately dwell on powers with a zero base and a fractional exponent. An expression of the form 0 m n can be given the following meaning: if m n > 0, then 0 m n = 0 m n = 0; if m n< 0 нуль остается не определен. Таким образом, возведение нуля в дробную положительную степень приводит к нулю: 0 7 12 = 0 , 0 3 2 5 = 0 , 0 0 , 024 = 0 , а в целую отрицательную - значения не имеет: 0 - 4 3 .

How to raise a number to an irrational power

The need to calculate the value of a power whose exponent is an irrational number does not arise so often. In practice, the task is usually limited to calculating an approximate value (up to a certain number of decimal places). This is usually calculated on a computer due to the complexity of such calculations, so we will not dwell on this in detail, we will only indicate the main provisions.

If we need to calculate the value of a power a with an irrational exponent a, then we take the decimal approximation of the exponent and count from it. The result will be an approximate answer. The more accurate the decimal approximation is, the more accurate the answer. Let's show with an example:

Example 11

Calculate the approximation of 2 to the power of 1.174367....

Solution

Let us limit ourselves to the decimal approximation a n = 1, 17. Let's carry out calculations using this number: 2 1, 17 ≈ 2, 250116. If we take, for example, the approximation a n = 1, 1743, then the answer will be a little more accurate: 2 1, 174367. . . ≈ 2 1, 1743 ≈ 2, 256833.

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If we ignore the eighth power, what do we see here? Let's remember the 7th grade program. So, do you remember? This is the formula for abbreviated multiplication, namely the difference of squares! We get:

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

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

Magically the terms changed places. This “phenomenon” applies to any expression to an even degree: we can easily change the signs in parentheses.

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 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, let us ask ourselves: why is this so?

Let's consider some degree with a base. Take, for example, and multiply by:

So, we multiplied the number by, and we got the same thing as it was - . What number should you multiply 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 will still get zero, this is clear. But on the other hand, like any number to the zero power, it must be equal. So how much of this is true? The mathematicians decided not to get involved and refused to raise zero to the zero power. That is, now we cannot not only divide by zero, but also raise it to the zero power.

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

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

Now let’s extend the resulting rule to an arbitrary degree:

So, let's formulate a rule:

A number with a negative power is the reciprocal of the same number with a positive power. But at the same time The base cannot be null:(because you can’t divide by).

Let's summarize:

I. The expression is not defined in the case. If, then.

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

III. A number 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 independent solutions:

Analysis of problems for independent solution:

I know, I know, the numbers are scary, but on the Unified State Exam you have to be prepared for anything! Solve these examples or analyze their solutions if you couldn’t solve them and you will learn to cope with them easily in the exam!

Let's continue to expand the range of numbers “suitable” as an exponent.

Now let's consider rational numbers. What numbers are called rational?

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

To understand what it is "fractional degree", consider the fraction:

Let's raise both sides of the equation to a 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 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 to.

That is, the root of the th power is the inverse operation of raising to a power: .

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

Now we add the numerator: what is it? The answer is easy to obtain using the power-to-power rule:

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

None!

Let us remember the rule: any number raised to an even power is a positive number. That is, it is impossible to extract even roots 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 the expression?

But here a problem arises.

A number can be represented as other, reducible fractions, for example, or.

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

Or another example: once, then you can write it down. But if we write down the indicator differently, we will again get into trouble: (that is, we got a completely different result!).

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

So if:

- natural number;
- integer;

Examples:

Rational exponents are very useful for transforming expressions with roots, for example:

5 examples to practice

Analysis of 5 examples for training

1. Don't forget about the usual properties of degrees:

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

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

Well, now comes the hardest part. Now we'll figure it out degree with irrational exponent.

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 natural, integer and rational exponents, each time we created a certain “image”, “analogy”, or description in more familiar terms.

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

...number to the zeroth power- this is, as it were, a number multiplied by itself once, that is, they have not yet begun to multiply it, which means that the number itself has not even appeared yet - therefore the result is only a certain “blank number”, namely a number;

...negative integer degree- it’s as if some “reverse process” had occurred, that is, the number was not multiplied by itself, but divided.

By the way, in science a degree with a complex exponent is often used, that is, the 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 to solve such examples :))

For example:

Decide for yourself:

Analysis of solutions:

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

Now look at the indicator. Doesn't he remind you of anything? Let us recall the formula for abbreviated multiplication of difference of squares:

In this case,

It turns out that:

Answer: .

2. We reduce fractions in exponents to the same form: either both decimals or both ordinary ones. We get, for example:

Answer: 16

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

ADVANCED LEVEL

Determination of degree

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

— degree base;
- exponent.

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

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

Degree with an integer exponent (0, ±1, ±2,...)

If the exponent is positive integer number:

Construction to the zero degree:

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 negative integer number:

(because you can’t divide by).

Once again about zeros: the expression is not defined in the case. If, then.

Examples:

Power with rational exponent

- natural number;
- integer;

Examples:

Properties of degrees

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?

A-priory:

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

But by definition it 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 there must be the same reasons. Therefore, we combine the powers with the base, but it remains a separate factor:

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

Under no circumstances can you write that.

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

Let's regroup this work like this:

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

In essence, this can be called “taking the indicator out of brackets.” But you can 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.

Power with a negative base.

Up to this point we have only discussed what it should be like index degrees. But what should be the basis? In powers of natural indicator the basis may 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, is the number positive or negative? A? ?

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

But the negative ones are a little more interesting. We remember the simple rule from 6th grade: “minus for 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. The following simple rules can be formulated:

even degree, - number positive.
Negative number raised to odd degree, - number negative.
A positive number to any degree is a positive number.
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: after all, it doesn’t matter what the base is equal to - the degree is even, which means the result will always be positive. Well, except when the base is zero. The base 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 we remember that, it becomes clear that, which means 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 by each other, divide them into pairs and get:

Before we look at the last rule, let's solve a few examples.

Calculate the expressions:

Solutions :

If we ignore the eighth power, what do we see here? Let's remember the 7th grade program. So, do you remember? This is the formula for abbreviated multiplication, namely the difference of squares!

We get:

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

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

Magically the terms changed places. This “phenomenon” applies to any expression to an even degree: we can easily change the signs in parentheses. But it's important to remember: All signs change at the same time! You can’t replace it with by changing only one disadvantage we don’t like!

Let's go back to the example:

And again the formula:

So now the last rule:

How will we prove it? Of course, as usual: let’s expand on the concept of degree and simplify it:

Well, now let's open the brackets. How many letters are there in total? times by multipliers - what does this remind you of? This is nothing more than a definition of an operation multiplication: There were only multipliers there. That is, this, by definition, is a power of a number with an exponent:

Example:

Degree with irrational exponent

In addition to information about degrees for the average level, we will analyze the degree with an irrational exponent. 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 numbers).

When studying degrees with natural, integer and rational exponents, each time we created a certain “image”, “analogy”, or description in more familiar terms. For example, a degree with a natural exponent is a number multiplied by itself several times; a number to the zero power is, as it were, a number multiplied by itself once, that is, they have not yet begun to multiply it, which means that the number itself has not even appeared yet - therefore the result is only a certain “blank number”, namely a number; a degree with an integer negative exponent - it’s as if some “reverse process” had 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). It is rather a purely mathematical object that mathematicians created to extend the concept of degree to the entire space of numbers.

By the way, in science a degree with a complex exponent is often used, that is, the 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:

Answers:

Let's remember the difference of squares formula. Answer: .
We reduce the fractions to the same form: either both decimals or both ordinary ones. We get, for example: .
Nothing special, we use the usual properties of degrees: