Degree and its properties. The Comprehensive Guide (2019)

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 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 .

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 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 .

Degree of

So, let's figure out what a power of a number is. To write the product of a number by itself, the abbreviated notation is used several times. So, instead of the product of six identical factors 4. 4 . 4 . 4 . 4 . 4 is written 4 6 and pronounced “four to the sixth power.”
4 . 4 . 4 . 4 . 4 . 4 = 4 6

The expression 4 6 is called a power of a number, where:
. 4 - base degree;
. 6 is the exponent.

In general, a degree with a base “a” and exponent “n” is written using the expression:

• A power of a number "a" with a natural exponent "n" greater than 1 is the product of "n" identical factors, each of which is equal to the number "a".

The entry a n reads like this: “a to the power of n” or “nth power of a.”

The exceptions are the following entries:
. a 2 - it can be pronounced as “a squared”;
. a 3 - it can be pronounced as “a cubed.”

• The power of the number “a” with exponent n = 1 is this number itself:
• a 1 = a
• Any number to the zero power is equal to one.
• a 0 = 1
• Zero to any natural power is equal to zero.
• 0 n = 0
• One to any power is equal to 1.
• 1 n = 1

The expression 0 0 (zero to the power of zero) is considered meaningless.
. (-32) 0 = 1
. 0 234 = 0
. 1 4 = 1
When solving examples, you need to remember that raising to a power means finding the value of a power.

Example. Raise to a power.
. 5 3 = 5 . 5 . 5 = 125
. 2.5 2 = 2.5 . 2.5 = 6.25
. (3 ) 4 = 3. 3. 3. 3 = 81
4 4 4 4 4 256

Raising a negative number to the power
The base (the number that is raised to the power) can be any number - positive, negative or zero.

• Raising a positive number to a power produces a positive number.

When zero is raised to a natural power, the result is zero.
When a negative number is raised to a power, the result can be either a positive number or a negative number. It depends on whether the exponent was an even or odd number.

Let's look at examples of raising negative numbers to powers.


From the examples considered, it is clear that if a negative number is raised to an odd power, then a negative number is obtained. Since the product of an odd number of negative factors is negative.

If a negative number is raised to an even power, it becomes a positive number. Since the product of an even number of negative factors is positive.

A negative number raised to an even power is a positive number.

• A negative number raised to an odd power is a negative number.
• The square of any number is a positive number or zero, that is:
• a 2 ≥ 0 for any a.

2 . (- 3) 2 = 2 . (- 3) . (- 3) = 2 . 9 = 18
. - 5 . (- 2) 3 = - 5 . (- 8) = 40

Note!
When solving examples of exponentiation, mistakes are often made, forgetting that the notations (- 5) 4 and -5 4 are different expressions. The results of raising these expressions to powers will be different.

Calculate (- 5) 4 means to find the value of the fourth power of a negative number.
(- 5) 4 = (- 5) . (- 5) . (- 5) . (- 5) = 625

While finding -5 4 means that the example needs to be solved in 2 steps:
1. Raise the positive number 5 to the fourth power.
5 4 = 5 . 5 . 5 . 5 = 625
2. Place a minus sign in front of the result obtained (that is, perform a subtraction action).
-5 4 = - 625
Example. Calculate: - 6 2 - (- 1) 4
- 6 2 - (- 1) 4 = - 37

1. 6 2 = 6 . 6 = 36
2. -6 2 = - 36
3. (- 1) 4 = (- 1) . (- 1) . (- 1) . (- 1) = 1
4. - (- 1) 4 = - 1
5. - 36 - 1 = - 37

Procedure in examples with degrees
Calculating a value is called the action of exponentiation. This is the third stage action.

• In expressions with powers that do not contain parentheses, exponentiation is performed first, then multiplication and division, and finally addition and subtraction.
• If the expression contains parentheses, then first perform the actions in the parentheses in the order indicated above, and then perform the remaining actions in the same order from left to right.

Example. Calculate:

Properties of degree

A power with a natural exponent has several important properties that allow us to simplify calculations in examples with powers.
Property No. 1
Product of powers

• When multiplying powers with the same bases, the base remains unchanged, and the exponents of the powers are added.
• a m. a n = a m+n , where a is any number, and m, n are any natural numbers.

This property of powers also applies to the product of three or more powers.
Examples.
. Simplify the expression.
b. b 2 . b 3. b 4. b 5 = b 1+2+3+4+5 = b 15

6 15 . 36 = 6 15 . 6 2 = 6 15+2 = 6 17

Present it as a degree.
(0,8) 3 . (0,8) 12 = (0,8) 3+12 = (0,8) 15

• Please note that in the indicated property we were talking only about the multiplication of powers with the same bases. It does not apply to their addition.
• You cannot replace the sum (3 3 + 3 2) with 3 3. This is understandable if you count 3 3 = 27 and 3 2 = 9; 27 + 9 = 36, and 3 5 = 243

Property No. 2
Partial degrees

• When dividing powers with the same bases, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.
• a m. a n = a m-n, where a is any number not equal to zero, and m, n are any natural numbers such that m > n.

Examples.
. Write the quotient as a power
(2b) 5: (2b) 3 = (2b) 5-3 = (2b) 2

Example. Solve the equation. We use the property of quotient powers.
3 8: t = 3 4

t = 3 8: 3 4

t = 3 8-4

t = 3 4

Answer: t = 3 4 = 81

Using properties No. 1 and No. 2, you can easily simplify expressions and perform calculations.
. Example. Simplify the expression.
4 5m+6 . 4 m+2: 4 4m+3 = 4 5m+6+m+2: 4 4m + 3 = 4 6m + 8 - 4m - 3 = 4 2m + 5

Please note that in Property 2 we were only talking about dividing powers with the same bases.
You cannot replace the difference (4 3 - 4 2) with 4 1. This is understandable if you count 4 3 = 64 and 4 2 = 16; 64 - 16 = 48, and 4 1 = 4
Be careful!

Property No. 3
Raising a degree to a power

• When raising a degree to a power, the base of the degree remains unchanged, and the exponents are multiplied.
• (a n) m = a n . m, where a is any number, and m, n are any natural numbers.

Example.
(a 4) 6 = a 4 . 6 = a 24
. Example. Express 3 20 as a power with a base of 32.
By the property of raising a degree to a power It is known that when raised to a power, exponents are multiplied, which means:

Properties 4
Product power

• When a power is raised to a product power, each factor is raised to that power and the results are multiplied.
• (a . b) n = a n . b n , where a, b are any rational numbers; n - any natural number.

Example 1.

(6 . a 2 . b 3 . c) 2 = 6 2 . a 2 . 2. b 3. 2. from 1 . 2 = 36 a 4 . b 6. from 2

Example 2.

(- x 2 . y) 6 = ((- 1) 6 . x 2 . 6 . y 1 . 6) = x 12 . y 6

Please note that property No. 4, like other properties of degrees, is also applied in reverse order.
(a n . b n)= (a . b) n

That is, to multiply powers with the same exponents, you can multiply the bases, but leave the exponent unchanged.
. Example. Calculate.

2 4 . 5 4 = (2 . 5) 4 = 10 4 = 10 000

Example. Calculate.

0,5 16 . 2 16 = (0,5 . 2) 16 = 1

In more complex examples, there may be cases where multiplication and division must be performed over powers with different bases and different exponents. In this case, we advise you to do the following.
For example, 4 5. 3 2 = 4 3. 4 2. 3 2 = 4 3. (4 . 3) 2 = 64 . 12 2 = 64. 144 = 9216

An example of raising a decimal to a power.
4 21 . (-0,25) 20 = 4 . 4 20 . (-0,25) 20 = 4 . (4 . (-0,25)) 20 = 4 . (- 1) 20 = 4 . 1 = 4

Properties 5
Power of a quotient (fraction)

• To raise a quotient to a power, you can raise the dividend and the divisor separately to this power, and divide the first result by the second.
• (a: b) n = a n: b n, where a, b are any rational numbers, b ≠ 0, n is any natural number.

Example. Present the expression as a quotient of powers.
(5: 3) 12 = 5 12: 3 12

Raising a fraction to a power

• When raising a fraction to a power, you must raise both the numerator and the denominator to a power.

Examples of raising fractions to powers.

How to raise a mixed number to a power
To raise a mixed number to a power, we first get rid of the integer part, turning the mixed number into an improper fraction. After this, we raise both the numerator and the denominator to a power.
Example.

The formula for raising a fraction to a power is used both from left to right and from right to left, that is, in order to divide degrees into each other by the same exponents, you can divide one base by another, and leave the exponent unchanged.

Example. Find the meaning of the expression in a rational way.

Properties of degrees

third power of number

Alternative descriptions

Geometric body

Geometric figure

Vessel for distilling and boiling liquids

Math trio

Volumetric square

Regular polyhedron

The plant from which vat paint was extracted

Third degree (mathematical)

Hexagon

A special case of a prism

Volume measure

Log shape

Hexahedron

Correct hexagon

Table salt and zinc sulphide crystallize in the shape of this geometric figure.

This regular polyhedron has 6 faces

This regular polyhedron has 8 vertices

What geometric figure does the ancient sanctuary of the Kaaba have?

Body square on all sides

A geometric body whose three projections are all squares

Number multiplied three times

Unit in which cut timber is measured

One of the forms of covering log houses

Third degree (math.)

Hexahedron in a simple way

3D square

Regular hexahedron

Makes a deuce an eight

Right hexagon

Polyhedron

Measure of cut timber

Shape of the Kaaba sanctuary

Third degree for mathematician

Polyhedron with 8 vertices

Salt crystal shape

All its projections are squares

Volume measure for logs

Combining 6 squares

Possessor of six ribs

Third degree in mathematics

Possessor of twelve ribs

Distillation...

Correct hexagon

Geometric body, regular polyhedron

Vessel for distilling and boiling liquids

Regular polyhedron with six faces

M. distillation vessel, alembic, projectile for distilling liquids, esp. wine The cube can be glass, clay, copper, etc., of different sizes and types; it is tightly covered with a cap, and the distillation liquid goes in pairs into the throat, neck, and from there into the refrigerator, and flows into the receiver. geometer. a rectangular, equilateral body bounded by six equal squares: a die, or chest, which has four sides, a lid and a bottom of one measure, represents a cube. arithmetic product, from multiplying any number twice by itself: cube of 4. Blood-sucking cube, healing projectile, for cutting skin; banks. Cube of fat, kamch. seal skin, filled with the fat of sea animals and sewn up all around; Kutyr. Plant. cube, Indigo, from which cube paint is extracted. The cube will diminish. generally a unit of cubic measure; among excavators, cubic fathom. Take out the earth cubes. Plant. Picris hieracioides, wood goose. Cubic, belonging to a cube, related. Cubic iron, boiler iron, thick sheets. Vat paint, blue vegetable paint made from plants. cube, indigo. Kubovik novg. A canvas blue sundress, otherwise dyed or tanned, is called a work sundress, a verkhnik, a dubenik, a sandalnik. Cubic, -shaped, forming a cube, into a geometer. and arithmetic meaning Cubic box, number; root, a number from which, when multiplied twice by itself, a cube was formed; will be the cube root of 8. Cubic measure, thick, measure of thickness: the extension from point to point is measured by a linear measure, linear; plane, surface with a measure from line to line, from edge to edge, with a flat, square measure; and every flow or capacity between two planes is a measure of thickness, cubic, thick. Cuboid, blocky, cuboidal, -shaped, almost cubical, close to a cube in appearance, chesty. Chop something, divide, break into cubes, cubes. Cube sugar and pour into cubes. Cube the earth, break it into cubes with a drawing; do cubic calculations. Mountain salt is cubed, divided, disintegrated into cubes. Kubatura f. a cube equal in thickness to a given body, for example. ball

What geometric shape does the ancient sanctuary of the Kaaba have?

Please note that this section discusses the concept degrees with natural exponent only and zero.

The concept and properties of powers with rational exponents (with negative and fractional) will be discussed in lessons for grade 8.

So, let's figure out what a power of a number is. To write the product of a number by itself, the abbreviated notation is used several times.

Instead of the product of six identical factors 4 · 4 · 4 · 4 · 4 · 4, write 4 6 and say “four to the sixth power”.

4 4 4 4 4 4 = 4 6

The expression 4 6 is called a power of a number, where:

• 4 — degree base;
• 6 — exponent.

In general, a degree with a base “a” and exponent “n” is written using the expression:

Remember!

The power of a number “a” with a natural exponent “n” greater than 1 is the product of “n” identical factors, each of which is equal to the number “a”.

The entry “a n” reads like this: “a to the power of n” or “nth power of the number a”.

The exceptions are the following entries:

• a 2 - it can be pronounced as “a squared”;
• a 3 - it can be pronounced as “a cubed.”

• a 2 - “a to the second power”;
• a 3 - “a to the third power.”

Special cases arise if the exponent is equal to one or zero (n = 1; n = 0).

Remember!

The power of the number “a” with exponent n = 1 is this number itself:
a 1 = a

Any number to the zero power is equal to one.
a 0 = 1

Zero to any natural power is equal to zero.
0 n = 0

One to any power is equal to 1.
1 n = 1

Expression 0 0 ( zero to the zero power) are considered meaningless.

• (−32) 0 = 1
• 0 253 = 0
• 1 4 = 1

When solving examples, you need to remember that raising to a power is finding a numerical or letter value after raising it to a power.

Example. Raise to a power.

• 5 3 = 5 5 5 = 125
• 2.5 2 = 2.5 2.5 = 6.25
• ( · = = 81

  256

Raising a negative number to the power

The base (the number that is raised to the power) can be any number - positive, negative or zero.

Remember!

Raising a positive number to a power produces a positive number.

When zero is raised to a natural power, the result is zero.

When a negative number is raised to a power, the result can be either a positive number or a negative number. It depends on whether the exponent was an even or odd number.

Let's look at examples of raising negative numbers to powers.

From the examples considered, it is clear that if a negative number is raised to an odd power, then a negative number is obtained. Since the product of an odd number of negative factors is negative.

If a negative number is raised to an even power, it becomes a positive number. Since the product of an even number of negative factors is positive.

Remember!

A negative number raised to an even power is a positive number.

A negative number raised to an odd power is a negative number.

The square of any number is a positive number or zero, that is:

a 2 ≥ 0 for any a.

• 2 · (−3) 2 = 2 · (−3) · (−3) = 2 · 9 = 18
• −5 · (−2) 3 = −5 · (−8) = 40

Note!

When solving examples of exponentiation, mistakes are often made, forgetting that the entries (−5) 4 and −5 4 are different expressions. The results of raising these expressions to powers will be different.

Calculating (−5) 4 means finding the value of the fourth power of a negative number.

(−5) 4 = (−5) · (−5) · (−5) · (−5) = 625

While finding “−5 4” means that the example needs to be solved in 2 steps:

1. Raise the positive number 5 to the fourth power.
   5 4 = 5 5 5 5 = 625
2. Place a minus sign in front of the result obtained (that is, perform a subtraction action).
   −5 4 = −625

Example. Calculate: −6 2 − (−1) 4

−6 2 − (−1) 4 = −37

1. 6 2 = 6 6 = 36
2. −6 2 = −36
3. (−1) 4 = (−1) · (−1) · (−1) · (−1) = 1
4. −(−1) 4 = −1
5. −36 − 1 = −37

Procedure in examples with degrees

Calculating a value is called the action of exponentiation. This is the third stage action.

Remember!

In expressions with powers that do not contain parentheses, first do exponentiation, then multiplication and division, and at the end addition and subtraction.

If the expression contains parentheses, then first perform the actions in the parentheses in the order indicated above, and then perform the remaining actions in the same order from left to right.

Example. Calculate:

To make it easier to solve examples, it is useful to know and use the table of degrees, which you can download for free on our website.

To check your results, you can use the calculator on our website "