The calculator helps you quickly raise a number to a power online. The base of the degree can be any number (both integers and reals). The exponent can also be an integer or real, and can also be positive or negative. Keep in mind that for negative numbers, raising to a non-integer power is undefined, so the calculator will report an error if you attempt it.
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What is a natural power of a number?
The number p is called the nth power of a number if p is equal to the number a multiplied by itself n times: p = a n = a·...·a
n - called exponent, and the number a is degree basis.
How to raise a number to a natural power?
To understand how to raise various numbers to natural powers, consider a few examples:
Example 1. Raise the number three to the fourth power. That is, it is necessary to calculate 3 4
Solution: as mentioned above, 3 4 = 3·3·3·3 = 81.
Answer: 3 4 = 81 .
Example 2. Raise the number five to the fifth power. That is, it is necessary to calculate 5 5
Solution: similarly, 5 5 = 5·5·5·5·5 = 3125.
Answer: 5 5 = 3125 .
Thus, to raise a number to a natural power, you just need to multiply it by itself n times.
What is a negative power of a number?
The negative power -n of a is one divided by a to the power of n: a -n = .In this case, a negative power exists only for non-zero numbers, since otherwise division by zero would occur.
How to raise a number to a negative integer power?
To raise a non-zero number to a negative power, you need to calculate the value of this number to the same positive power and divide one by the result.
Example 1. Raise the number two to the negative fourth power. That is, you need to calculate 2 -4
Solution: as stated above, 2 -4 = = = 0.0625.Answer: 2 -4 = 0.0625 .
In this article we will figure out what it is degree of. Here we will give definitions of the power of a number, while we will consider in detail all possible exponents, starting with the natural exponent and ending with the irrational one. In the material you will find a lot of examples of degrees, covering all the subtleties that arise.
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Power with natural exponent, square of a number, cube of a number
Let's start with . Looking ahead, let's say that the definition of the power of a number a with natural exponent n is given for a, which we will call degree basis, and n, which we will call exponent. We also note that a degree with a natural exponent is determined through a product, so to understand the material below you need to have an understanding of multiplying numbers.
Definition.
Power of a number with natural exponent n is an expression of the form a n, the value of which is equal to the product of n factors, each of which is equal to a, that is, .
In particular, the power of a number a with exponent 1 is the number a itself, that is, a 1 =a.
It’s worth mentioning right away about the rules for reading degrees. The universal way to read the notation a n is: “a to the power of n”. In some cases, the following options are also acceptable: “a to the nth power” and “nth power of a”. For example, let's take the power 8 12, this is “eight to the power of twelve”, or “eight to the twelfth power”, or “twelfth power of eight”.
The second power of a number, as well as the third power of a number, have their own names. The second power of a number is called square the number, for example, 7 2 is read as “seven squared” or “the square of the number seven.” The third power of a number is called cubed numbers, for example, 5 3 can be read as “five cubed” or you can say “cube of the number 5”.
It's time to bring examples of degrees with natural exponents. Let's start with the degree 5 7, here 5 is the base of the degree, and 7 is the exponent. Let's give another example: 4.32 is the base, and the natural number 9 is the exponent (4.32) 9 .
Please note that in the last example, the base of the power 4.32 is written in parentheses: to avoid discrepancies, we will put in parentheses all bases of the power that are different from natural numbers. As an example, we give the following degrees with natural exponents 
, their bases are not natural numbers, so they are written in parentheses. Well, for complete clarity, at this point we will show the difference contained in records of the form (−2) 3 and −2 3. The expression (−2) 3 is a power of −2 with a natural exponent of 3, and the expression −2 3 (it can be written as −(2 3) ) corresponds to the number, the value of the power 2 3 .
Note that there is a notation for the power of a number a with an exponent n of the form a^n. Moreover, if n is a multi-valued natural number, then the exponent is taken in brackets. For example, 4^9 is another notation for the power of 4 9 . And here are some more examples of writing degrees using the symbol “^”: 14^(21) , (−2,1)^(155) . In what follows, we will primarily use degree notation of the form a n .
One of the problems inverse to raising to a power with a natural exponent is the problem of finding the base of a power from a known value of the power and a known exponent. This task leads to .
It is known that the set of rational numbers consists of integers and fractions, and each fraction can be represented as a positive or negative ordinary fraction. We defined a degree with an integer exponent in the previous paragraph, therefore, in order to complete the definition of a degree with a rational exponent, we need to give meaning to the degree of the number a with a fractional exponent m/n, where m is an integer and n is a natural number. Let's do it.
Let's consider a degree with a fractional exponent of the form . For the power-to-power property to remain valid, the equality must hold 
. If we take into account the resulting equality and how we determined , then it is logical to accept it provided that for given m, n and a the expression makes sense.
It is easy to check that for all properties of a degree with an integer exponent are valid (this was done in the section properties of a degree with a rational exponent).
The above reasoning allows us to make the following conclusion: if given m, n and a the expression makes sense, then the power of a with a fractional exponent m/n is called the nth root of a to the power of m.
This statement brings us close to the definition of a degree with a fractional exponent. All that remains is to describe at what m, n and a the expression makes sense. Depending on the restrictions placed on m, n and a, there are two main approaches.
The easiest way is to impose a constraint on a by taking a≥0 for positive m and a>0 for negative m (since for m≤0 the degree 0 of m is not defined). Then we get the following definition of a degree with a fractional exponent.
Definition.
Power of a positive number a with fractional exponent m/n, where m is an integer and n is a natural number, is called the nth root of the number a to the power of m, that is, .
The fractional power of zero is also determined with the only caveat that the indicator must be positive.
Definition.
Power of zero with fractional positive exponent m/n, where m is a positive integer and n is a natural number, is defined as 
.
When the degree is not determined, that is, the degree of the number zero with a fractional negative exponent does not make sense.
It should be noted that with this definition of a degree with a fractional exponent, there is one caveat: for some negative a and some m and n, the expression makes sense, and we discarded these cases by introducing the condition a≥0. For example, the entries make sense 
or , and the definition given above forces us to say that powers with a fractional exponent of the form 
do not make sense, since the base should not be negative.
Another approach to determining a degree with a fractional exponent m/n is to separately consider even and odd exponents of the root. This approach requires an additional condition: the power of the number a, the exponent of which is , is considered to be the power of the number a, the exponent of which is the corresponding irreducible fraction (we will explain the importance of this condition below). That is, if m/n is an irreducible fraction, then for any natural number k the degree is first replaced by .
For even n and positive m, the expression makes sense for any non-negative a (an even root of a negative number does not make sense); for negative m, the number a must still be different from zero (otherwise there will be division by zero). And for odd n and positive m, the number a can be any (the root of an odd degree is defined for any real number), and for negative m, the number a must be different from zero (so that there is no division by zero).
The above reasoning leads us to this definition of a degree with a fractional exponent.
Definition.
Let m/n be an irreducible fraction, m an integer, and n a natural number. For any reducible fraction, the degree is replaced by . The power of a number with an irreducible fractional exponent m/n is for
Let us explain why a degree with a reducible fractional exponent is first replaced by a degree with an irreducible exponent. If we simply defined the degree as , and did not make a reservation about the irreducibility of the fraction m/n, then we would be faced with situations similar to the following: since 6/10 = 3/5, then the equality must hold 
, But 
, A .
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 power.
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 undefined 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 down 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 removed entirely, or the numerator can be reduced to one, without changing the meaning of the expression.
So, a/b = 1/ba -1 , or ab -1 .
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.
Yandex.RTB R-A-339285-1
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 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 transform the value of the indicator into an ordinary 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 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 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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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 or 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.
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.
Bottom line
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