Continuing the conversation about the power of a number, it is logical to figure out how to find the value of the power. This process is called exponentiation. In this article we will study how exponentiation is performed, while we will touch on all possible exponents - natural, integer, rational and irrational. And according to tradition, we will consider in detail solutions to examples of raising numbers to various powers.
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What does "exponentiation" mean?
Let's start by explaining what is called exponentiation. Here is the relevant definition.
Definition.
Exponentiation- this is finding the value of the power of a number.
Thus, finding the value of the power of a number a with exponent r and raising the number a to the power r are 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 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 a number a to a fractional power m/n, first the nth root of the number a is taken, after which the resulting result is raised to an integer power m.
Let's look at solutions to examples of raising to a fractional power.
Example.
Calculate the value of the degree.
Solution.
We will show two solutions.
First way. By definition of a degree with a fractional exponent. We calculate the value of the degree under the root sign, and then extract the cube root: 
.
Second way. By the definition of a degree with a fractional exponent and based on the properties of the roots, the following equalities are true: 
. Now we extract the root 
, finally, we raise it to an integer power 
.
Obviously, the obtained results of raising to a fractional power coincide.
Answer:
Note that a fractional exponent can be written as a decimal fraction or a mixed number, in these cases it should be replaced with the corresponding ordinary fraction, and then raised to a power.
Example.
Calculate (44.89) 2.5.
Solution.
Let's write the exponent in the form of an ordinary fraction (if necessary, see the article): 
. Now we perform the 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 labor-intensive process (especially when the numerator and denominator of the fractional exponent contain sufficiently large numbers), which is usually carried out using computer technology.
To conclude this point, let us dwell on raising the number zero to a fractional power. We gave the following meaning to the fractional power of zero of the form: when we have 
, and at zero to the m/n power is not defined. So, zero to a fractional positive power is zero, for example, 
. And zero in a fractional negative power does not make sense, for example, the expressions 0 -4.3 do not make sense.
Raising to an irrational power
Sometimes it becomes necessary to find out the value of the power of a number with an irrational exponent. At the same time, in practical purposes Usually it is enough to obtain the value of the degree up to a certain sign. Let us immediately note that this value in practice is calculated using electronic computer technology, since raising it to ir rational degree manually requires large quantity cumbersome calculations. But still we will describe in general outline the essence of the action.
To obtain an approximate value of the power of a number a with an irrational exponent, some decimal approximation of the exponent is taken and the value of the power is calculated. This value is an approximate value of the power of the number a with an irrational exponent. The more accurate the decimal approximation of a number is taken initially, the more exact value degree will be obtained 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 the irrational exponent: . Now we raise 2 to the rational power 1.17 (we described the essence of this process in the previous paragraph), we get 2 1.17 ≈2.250116. Thus, 2 1,174367... ≈2 1,17 ≈2,250116 . If we take a more accurate decimal approximation of the irrational exponent, for example, then we obtain a more accurate value of the original exponent: 2 1,174367... ≈2 1,1743 ≈2,256833 .
Bibliography.
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics textbook for 5th grade. educational institutions.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 7th grade. educational institutions.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8th grade. educational institutions.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 9th grade. educational institutions.
- Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the beginnings of analysis: Textbook for grades 10 - 11 of general education institutions.
- Gusev V.A., Mordkovich A.G. Mathematics (a manual for those entering technical schools).
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 at the most simple 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 considering the issue at 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 degree can be either 3 or 4 or any number to infinity, and this number determines the degree 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 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.
Anyway the best option 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.
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.
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 solving arithmetic and algebraic problems, it is sometimes necessary to construct fraction V square. The easiest way to do this is when fraction decimal - a regular calculator is enough. However, if fraction ordinary or mixed, then when raising such a number to square Some difficulties may arise.
You will need
- calculator, computer, Excel application.
Instructions
To raise a decimal fraction V square, take an engineering one, type on it what is being built in square fraction and press the raise to the second power key. On most calculators this button is labeled "x²". On a standard Windows calculator, the function of raising to square looks like "x^2". For example, square the decimal fraction 3.14 will be equal to: 3.14² = 9.8596.
To build into square decimal fraction on a regular (accounting) calculator, multiply this number by itself. By the way, some models of calculators provide the ability to raise a number to square even in the absence of a special button. Therefore, first read the instructions for your specific calculator. Sometimes "tricky" exponentiations are given on the back cover or on the calculator. For example, on many calculators, to raise a number to square Just press the “x” and “=” buttons.
For construction in square common fraction (consisting of a numerator and a denominator), raise to square separately the numerator and denominator of this fraction. That is, use the following rule: (h / z)² = h² / z², where h is the numerator of the fraction, z is the denominator of the fraction. Example: (3/4)² = 3²/4² = 9/16.
If being built in square fraction– mixed (consists of an integer part and an ordinary fraction), then first reduce it to an ordinary form. That is, apply the following formula: (c h/z)² = ((c*z+ch) / z)² = (c*z+ch)² / z², where c is the integer part of the mixed fraction. Example: (3 2/5)² = ((3*5+2) / 5)² = (3*5+2)² / 5² = 17² / 5² = 289/25 = 11 14/25.
If in square(not ) fractions happen all the time, then use MS Excel. To do this, enter the following formula into one of the tables: = DEGREE (A2;2) where A2 is the address of the cell into which the raised value will be entered square fraction.To tell the program that the input number should be treated as fraction yu (i.e. do not convert it to decimal), type before fraction I have the number “0” and the sign “space”. That is, to enter, for example, the fraction 2/3, you need to enter: “0 2/3” (and press Enter). In this case, the decimal representation of the entered fraction will be displayed in the input line. The value and representation of the fraction itself will be saved in its original form. In addition, when using mathematical functions whose arguments are ordinary fractions, the result will also be presented as an ordinary fraction. Hence square the fraction 2/3 will be represented as 4/9.
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.
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The concept of exponentiation
Let's start by formulating basic definitions.
Definition 1
Exponentiation- this 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. This 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 3 to the power - 2.
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 decimal. 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 transform the value of the indicator into common fraction - 44 , 89 2 , 5 = 49 , 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 indicators- enough hard work. 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 the power whose exponent is irrational number, does not occur very 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 approximate value of 21, 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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The lesson will look at a more generalized version of multiplying fractions - raising to a power. First of all, we will talk about natural powers of fractions and examples that demonstrate similar operations with fractions. At the beginning of the lesson, we will also review raising whole expressions to natural powers and see how this will be useful for solving further examples.
Topic: Algebraic fractions. Arithmetic operations on algebraic fractions
Lesson: Raising an algebraic fraction to a power
1. Rules for raising fractions and whole expressions to natural powers with elementary examples
The rule for raising ordinary and algebraic fractions to a natural power:
You can draw an analogy with the degree of an entire expression and remember what is meant by raising it to a power:
Example 1. 
.
As can be seen from the example, raising a fraction to a power is special case multiplying fractions, which was studied in the previous lesson.
Example 2. a) , b) 
- the minus goes away, because we raised the expression to an even power.
For the convenience of working with degrees, let us recall the basic rules for raising to a natural degree:
- product of powers;
- division of degrees;
Raising a degree to a degree;
Degree of product.
Example 3. - we know this from the topic “Exponentiation of whole expressions”, except for one case: it does not exist.
2. The simplest examples of raising algebraic fractions to natural powers
Example 4. Raise a fraction to a power.
Solution. When raised to an even power, the minus goes away:
Example 5. Raise a fraction to a power.
Solution. Now we use the rules for raising a degree to a power immediately without a separate schedule:
.
Now let's look at combined problems in which we will need to raise fractions to powers, multiply them, and divide them.
Example 6. Perform actions.
Solution. 
. Next you need to make a reduction. Let us describe once in detail how we will do this, and then we will indicate the result immediately by analogy: . Similarly (or according to the rule of division of powers). We have: .
Example 7. Perform actions.
Solution. . The reduction was carried out by analogy with the example discussed earlier.
Example 8. Perform actions.
Solution. . IN in this example We once again described in more detail the process of reducing powers in fractions to consolidate this method.
3. More complex examples for raising algebraic fractions to natural powers (taking into account signs and with terms in brackets)
Example 9: Perform actions 
.
Solution. In this example, we will already skip the separate multiplication of fractions, and immediately use the rule for multiplying them and write them under one denominator. At the same time, we follow the signs - in this case, the fractions are raised to even powers, so the minuses disappear. At the end we will perform the reduction.
Example 10: Perform actions 
.
Solution. In this example, there is division of fractions; remember that in this case the first fraction is multiplied by the second, but inverted.