can be found using multiplication. For example: 5+5+5+5+5+5=5x6. They say about such an expression that the sum of equal terms has been folded into a product. And vice versa, if we read this equality from right to left, we get that we have expanded the sum of equal terms. Similarly, you can fold the product of several equal factors 5x5x5x5x5x5=5 6 .

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

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

5 - base of degree;

6 - exponent.

The operations by which the product of equal factors is folded into a power are called exponentiation.

In general, a power with base "a" and exponent "n" is written as

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

If the base of the degree "a" is 1, then the value of the degree for any natural n will be equal to 1. For example, 1 5 \u003d 1, 1 256 \u003d 1

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

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

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

Any number can be raised to a power - positive, negative or zero. However, the following rules are not used:

When finding the degree of a positive number, a positive number is obtained.

When calculating zero in kind, we get zero.

x m х n = x m + n

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

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

x m / x n \u003d x m - n , where, m > n

ex: 13 3.8 / 13 -0.2 = 13 (3.8 -0.2) = 13 3.6

When calculating exponentiation We do not change the base, but we multiply the exponents by each other.

(at m )n = y m n

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

(X · y) n = x n · m ,

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

When performing calculations for exponentiation of a fraction we raise the numerator and denominator of the fraction to the given power

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

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

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

When calculating expressions without brackets, but containing powers, first of all, exponentiation is performed, then multiplication and division, and only then addition and subtraction.

If it is necessary to evaluate an expression containing brackets, then first, in the order indicated above, we do the calculations in brackets, and then the remaining actions in the same order from left to right.

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

The degree is also generalized to the case of an arbitrary (rational or irrational, as well as complex) exponent.

Big Encyclopedic Dictionary. 2000 .

Synonyms:

See what "DEGREE" is in other dictionaries:

Degrees, pl. degrees, degrees, wives. 1. Comparative value, comparative quantity, comparative size, comparative quality of what n. degree of culture. High degree of skill. Degree of relationship (number of births linking ... ... Explanatory Dictionary of Ushakov

Female step, row, category, order, from cases by quality, dignity; the place and the very assembly of the homogeneous, equal in everything, where the ladder order, ascending and descending, is supposed. The kingdom of fossils, plants and animals, these are the three degrees ... ... Dahl's Explanatory Dictionary

Step, category, row, stage, phase, height, point, degree, level, ordinary, dignity, rank, rank. Sequence of degrees ladder, hierarchy. Educational, property qualification. The case has entered a new phase. Consumption in the last degree ... Synonym dictionary

DEGREE, and, pl. and, she, wives. 1. Measure, the comparative value of which n. C. preparedness. C. pollution. 2. The same as the title (in 1 meaning), as well as (obsolete) rank, rank. Scientist s. Doctor of Sciences. Reach high levels. 3. usually with order. number… … Explanatory dictionary of Ozhegov

degree- degree of dissociation degree of oxidation degree of absorption ... Chemical terms

- (power) An indicator indicating a certain number of multiplications of a number by itself, nth power of x means x; multiplied by itself n times; n is the exponent. Powers can be positive or negative: x n means that ... Economic dictionary

POWER, in mathematics, the result of multiplying a number or VARIABLE by itself a certain number of times. Thus, a2 (= a 3 a) is the second power of a; a3 third degree; a4 fourth, etc. The number to be multiplied (in this example, a) is called the base ... ... Scientific and technical encyclopedic dictionary

degree- degree, pl. degree, genus degrees (wrong degrees) ... Dictionary of pronunciation and stress difficulties in modern Russian

DEGREE- (1) dissociation is a value that characterizes the state of equilibrium of the reaction (see) in homogeneous (gaseous and liquid) systems; is expressed by the ratio of the number of molecules that have decayed (dissociated) into their constituent parts (atoms, molecules, nones), to ... ... Great Polytechnic Encyclopedia

The term "power" can mean: In mathematics Exponentiation Cartesian power Root of the nth degree Set power Polynomial power Differential equation power Mapping power Point power in geometry Powers of a thousand ... ... Wikipedia

Books

• Degree of trust, Vladimir Voinovich, "Degree of trust" - the first historical story by V. Voinovich. It is dedicated to the remarkable revolutionary Vera Nikolaevna Figner. The author focuses on the key points ... Series: Fiery Revolutionaries Publisher: Political Literature Publishing House,
• The degree of readiness of the business process management system for the introduction of information technologies (assessment method) , AV Kostrov , The article sets the task of assessing the degree of readiness of the business process management system for informatization. It is proposed to display verbal descriptions of the stages of maturity with a variety of particular ... Series: Applied Informatics. Science articles Publisher:

A product in which all factors are the same can be written shorter:

4 4 4 = 4 3

The expression 4 3 (and also the result of its evaluation) is called degree.

A degree is a shorthand notation for the product of like factors.

The number showing the number of identical factors is called exponent. The number raised to a power is called base of degree:

Record 4 3 reads like this: four to the power of three or four to the third power.

degree of number a with a natural indicator n(where n> 1) call the product n multipliers, each of which is equal to a.

Example 1 Calculate 2 4:

Example 2 Calculate 3 7:

If any number is taken by the factor 2 times, then the product is called the second power of this number, if any number is taken by the factor 3 times, then the product is called the third power of this number, etc. For example, the product of 16 from the first example is the fourth power of 2.

The first power of a number is the number itself. For example, 2 1 \u003d 2, 5 1 \u003d 5, 100 1 \u003d 100, i.e. the first power of any number is equal to the number itself:

a 1 = a

The second power of a number is called something else. square numbers. For example, writing 5 2 reads five squared. The third power of a number is called something else. cube numbers. For example, writing 5 3 reads five cubed. These names are borrowed from geometry.

This is the calculation of the degree value. For example, if the task is to calculate the value of the power of 3 5, then it can be reformulated as follows: raise the number 3 to the fifth power.

Example: calculate the value of the degree 3 5 .

Solution: this degree is equal to the product: 3 3 3 3 3. We multiply the factors and get the answer: 243.

Answer: 3 5 = 243.

The degree is often used to write very large or very small numbers. For example, the speed of light, which is approximately equal to 300,000,000 (three hundred million) meters per second, is more convenient to write as follows: 3 · 10 8 m/s.

A degree can be used to represent a bit unit as a degree:

399 = 3 100 + 9 10 + 9 1 = 3 10 2 + 9 10 1 + 9 1

Also, the degree is often used in writing the decomposition of a number into prime factors:

1000 = 2 3 5 3

exponentiation calculator

This calculator will help you perform exponentiation. Just enter the base with the exponent and click the Calculate button.

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 is 10.
10.10.10 = 1000, third degree.
10.10.10.10 = 10000 fourth degree.

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

The original number is called root degrees of that number, because that is the number from which the degrees were created.

However, it is not very convenient, especially in the case of high powers, to write down all the factors that make up the powers. Therefore, an abbreviated notation method is used. The root of the degree is written only once, and to the right and a little higher next to 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 of a must be multiplied by itself twice to get the power of aa. Also, a 3 means aaa, that is, here a is repeated three times as a multiplier.

The exponent of the first power is 1, but it is usually not 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 whole. The exponent indicates how often the value is taken as factor in the work.
So, 4a = a + a + a + a. But a 4 = a.a.a.a

The exponential notation has the peculiar advantage of allowing us to express unknown degree. For this purpose, instead of a number, the exponent is written letter. In the process of solving the problem, we can get a value that, as we know, is some degree of another magnitude. But so far we do not know if 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 not defined what degree. So, b m and d n are raised to the powers of m and n. When the exponent is found, number substituted for 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 ​​with exponents is also a great 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 cubed, 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 factor or reduced 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 ;
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 on a, we will successfully get multiple values.

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

If we start left divide on 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 such a 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 full row will 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 values on right from unit is reverse values ​​to the left of one. Therefore, these degrees can be called inverse powers a. One can also say that the powers on the left are the inverse 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 a 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 inverse 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 to make the exponents a complete series with 1 as the total difference, a/a or 1 is considered as such that has no 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 separately taken degrees will have the form:
+4,+3,+2,+1,0,-1,-2,-3,-4.

The root of the 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 degrees 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. Exponentiation rule:

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

This rule is common to all examples that may arise in the process of exponentiation. But it will be correct to explain 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 the exponent indicates.

The fourth power 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 degree 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 degree of a product, we can either operate on the entire product at once, or we can operate on each factor separately, and then multiply their values ​​with degrees.

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

Example 2. The third power of 4b 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, the first power.
(a + b) 1 = a 2 + 2ab + b 2 , second power (a + b).
(a + b) 3 \u003d a 3 + 3a 2 b + 3ab 2 + b 3, third degree.
(a + b) 4 \u003d a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4, fourth degree.

Square a - b, there is a 2 - 2ab + b 2 .

The square 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 b + 2.

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

Exercise 4. Find the sixth degree 1 - b.

Sum squares amounts and difference binomials are so common 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. Hence, the square of the sum and difference of the binomials can be found using the following rule.

The square of a binomial, both 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 difference binomial 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 are 4a 2 + 4ab + b 2 .

Example 2. The square ab + cd is a 2 b 2 + 2abcd + c 2 d 2 .

Example 3. The square 3d - h 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 efficient to write degrees no multiplication.

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

In such cases, the brackets 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 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 product of their squares. But they are equal to each other.

The cube a.(b + d), is 3 , or a 3 .(b + d) 3 .

It is also necessary to take into account the sign in front of the members involved. It is very important to remember that when the root of a power is positive, all its positive powers are also positive. But when the root is negative, values ​​from odd powers are negative, while the values even degrees are positive.

The second power (- 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 exponent 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 value already 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 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 .

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 a 2 b -3 is a 8 b -12 or a 8 / b 12 .

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

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

However, it must be remembered here that if a sign previous degree is "-", then it should 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 multiplication signs, 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 of -a 3 is a 3n .

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

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

The square a/b is a 2 /b 2 . According to the rule of multiplication of 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 where one of the terms is a fraction.

1. Find the square 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 a + 2/3 is a 2 + 4a/3 + 4/9.

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

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

Previously, it was shown that fractional coefficient can be moved from the numerator to the denominator or from the denominator to the numerator. Using the scheme of writing inverse powers, it can be seen that any multiplier can also be moved if the sign of the degree is changed.

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

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

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

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

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

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

Please note that this section deals with the concept degrees only with a natural indicator and zero.

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

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

Instead of multiplying six identical factors 4 4 4 4 4 4 they write 4 6 and say "four to the sixth power."

4 4 4 4 4 4 = 4 6

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

• 4 — base of degree;
• 6 — exponent.

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

Remember!

The degree of the number "a" with a natural exponent" n",greater than 1, is the product" n»Identical factors, each of which is equal to the number "a".

The record " a n"It reads like this:" and to the power n "or" n-th power of the number a".

The exceptions are the entries:

• a 2 - it can be pronounced as “a squared”;
• a 3 - it can be pronounced as "a in a cube."

• a 2 - "and to the second degree";
• a 3 - "a to the third degree."

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

Remember!

The degree of the number "a" with the exponent n \u003d 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 equals 1.
1n=1

Expression 0 0 ( zero to zero power) is considered meaningless.

• (−32) 0 = 1
• 0 253 = 0
• 1 4 = 1

When solving examples, you need to remember that raising to a power is called finding a numeric or literal 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

Exponentiation of a negative number

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

Remember!

Raising a positive number to a power results in a positive number.

Raising zero to a natural power results in zero.

When raising a negative number 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.

Consider examples of raising negative numbers to a power.

It can be seen from the examples considered 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, then a positive number is obtained. 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 exponentiation examples, mistakes are often made, forgetting that the entries (−5) 4 and −5 4 are different expressions. The results of raising to a power of these expressions 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. Put 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 for Examples with Degrees

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

Remember!

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

If there are brackets in the expression, then first, in the order indicated above, the actions in the brackets are performed, and then the remaining actions in the same order from left to right.

Example. Calculate:

To facilitate the solution of examples, it is useful to know and use the degree table, which you can download for free on our website.

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