Power formulas used in the process of reduction and simplification complex expressions, in solving equations and inequalities.

Number c is n-th power of a number a When:

Operations with degrees.

1. Multiplying degrees with the same base, their indicators add up:

a ma n = a m + n .

2. In the division of degrees with the same base, their indicators are subtracted:

3. The degree of the product of 2 or more factors is equal to the product of the degrees of these factors:

(abc…) n = a n b n c n …

4. The degree of a fraction is equal to the ratio of the degrees of the dividend and the divisor:

(a/b) n = a n / b n .

5. Raising a power to a power, the exponents are multiplied:

(am) n = a m n .

Each formula above is correct in the directions from left to right and vice versa.

For example. (2 3 5/15)² = 2² 3² 5²/15² = 900/225 = 4.

Operations with roots.

1. The root of the product of several factors is equal to the product of the roots of these factors:

2. The root of the relationship is equal to the ratio divisible and divisor of roots:

3. When raising a root to a power, it is enough to raise the root number to this power:

4. If we increase the degree of the root in n once and at the same time raise to n th power is a root number, then the value of the root will not change:

5. If we decrease the degree of the root in n root at the same time n th degree from the radical number, then the value of the root will not change:

Degree with a negative exponent. The degree of a number with a non-positive (integer) exponent is defined as one divided by the degree of the same number with an exponent equal to the absolute value of the non-positive exponent:

Formula a m:a n = a m - n can be used not only for m> n, but also at m< n.

For example. a4:a 7 = a 4 - 7 = a -3.

To formula a m:a n = a m - n became fair at m=n, you need the presence of the zero degree.

Degree with zero exponent. The power of any non-zero number with a zero exponent is equal to one.

For example. 2 0 = 1,(-5) 0 = 1,(-3/5) 0 = 1.

Degree with a fractional exponent. To raise a real number A to a degree m/n, you need to extract the root n th degree of m th power of this number A.

Each arithmetic operation sometimes becomes too cumbersome to record and they try to simplify it. It used to be the same with the addition operation. It was necessary for people to carry out repeated additions of the same type, for example, to calculate the cost of one hundred Persian carpets, the cost of which is 3 gold coins for each. 3 + 3 + 3 + ... + 3 = 300. Due to the bulkiness, it was thought to reduce the notation to 3 * 100 = 300. In fact, the notation “three times one hundred” means that you need to take one hundred triples and add them together. Multiplication took root, gained general popularity. But the world does not stand still, and in the Middle Ages it became necessary to carry out repeated multiplication of the same type. I recall an old Indian riddle about a sage who asked for wheat grains in the following quantity as a reward for the work done: for the first cell of the chessboard he asked for one grain, for the second - two, the third - four, the fifth - eight, and so on. This is how the first multiplication of powers appeared, because the number of grains was equal to two to the power of the cell number. For example, on the last cell there would be 2*2*2*…*2 = 2^63 grains, which is equal to a number 18 characters long, which, in fact, is the meaning of the riddle.

The operation of raising to a power took root quite quickly, and it also quickly became necessary to carry out addition, subtraction, division and multiplication of degrees. The latter is worth considering in more detail. The formulas for adding powers are simple and easy to remember. In addition, it is very easy to understand where they come from if the power operation is replaced by multiplication. But first you need to understand the elementary terminology. The expression a ^ b (read "a to the power of b") means that the number a should be multiplied by itself b times, and "a" is called the base of the degree, and "b" is the exponent. If the bases of the powers are the same, then the formulas are derived quite simply. Specific example: find the value of the expression 2^3 * 2^4. To know what should happen, you should find out the answer on the computer before starting the solution. Having hammered this expression into any online calculator, a search engine, typing "multiplying degrees with different grounds and the same" or a mathematical package, the output will be 128. Now let's write this expression: 2^3 = 2*2*2, and 2^4 = 2*2*2*2. It turns out that 2^3 * 2^4 \u003d 2 * 2 * 2 * 2 * 2 * 2 * 2 \u003d 2 ^ 7 \u003d 2 ^ (3 + 4) ... It turns out that the product of powers with the same base is equal to the base raised to a power equal to the sum of the previous two powers.

You might think that this is an accident, but no: any other example can only confirm this rule. Thus, in general view the formula looks like this: a^n * a^m = a^(n+m) . There is also a rule that any number to the zero power is equal to one. Here we should remember the rule of negative powers: a^(-n) = 1 / a^n. That is, if 2^3 = 8, then 2^(-3) = 1/8. Using this rule, we can prove the equality a^0 = 1: a^0 = a^(n-n) = a^n * a^(-n) = a^(n) * 1/a^(n) , a^ (n) can be reduced and remains one. From this, the rule is derived that the quotient of powers with the same bases is equal to this base to a degree equal to the quotient of the dividend and divisor: a ^ n: a ^ m = a ^ (n-m) . Example: Simplify the expression 2^3 * 2^5 * 2^(-7) *2^0: 2^(-2) . Multiplication is a commutative operation, so the multiplication exponents must first be added: 2^3 * 2^5 * 2^(-7) *2^0 = 2^(3+5-7+0) = 2^1 =2. The next step is to deal with the division into negative degree. It is necessary to subtract the divisor exponent from the dividend exponent: 2^1: 2^(-2) = 2^(1-(-2)) = 2^(1+2) = 2^3 = 8. It turns out that the operation of dividing by a negative degree is identical to the operation of multiplication by a similar positive exponent. So the final answer is 8.

There are examples where non-canonical multiplication of powers takes place. Multiplying powers with different bases is very often much more difficult, and sometimes even impossible. Several examples of various possible approaches should be given. Example: simplify the expression 3^7 * 9^(-2) * 81^3 * 243^(-2) * 729. Obviously, there is a multiplication of powers with different bases. But, it should be noted that all bases are different powers of a triple. 9 = 3^2.1 = 3^4.3 = 3^5.9 = 3^6. Using the rule (a^n) ^m = a^(n*m) , you should rewrite the expression in a more convenient form: 3^7 * (3^2) ^(-2) * (3^4) ^3 * ( 3^5) ^(-2) * 3^6 = 3^7 * 3^(-4) * 3^(12) * 3^(-10) * 3^6 = 3^(7-4+12 -10+6) = 3^(11) . Answer: 3^11. In cases where various grounds, on equal performance the rule a^n * b^n = (a*b) ^n works. For example, 3^3 * 7^3 = 21^3. Otherwise, when there are different bases and indicators, it is impossible to make a full multiplication. Sometimes you can partially simplify or resort to the help of computer technology.

The concept of a degree in mathematics is introduced as early as the 7th grade in an algebra lesson. And in the future, throughout the course of studying mathematics, this concept is actively used in its various forms. Degrees are a rather difficult topic, requiring memorization of values ​​and the ability to correctly and quickly count. For faster and better work with mathematics degrees, they came up with the properties of a degree. They help to cut down on big calculations, to convert a huge example into a single number to some extent. There are not so many properties, and all of them are easy to remember and apply in practice. Therefore, the article discusses the main properties of the degree, as well as where they are applied.

degree properties

We will consider 12 properties of a degree, including properties of powers with the same base, and give an example for each property. Each of these properties will help you solve problems with degrees faster, as well as save you from numerous computational errors.

1st property.

Many people very often forget about this property, make mistakes, representing a number to the zero degree as zero.

2nd property.

3rd property.

It must be remembered that this property can only be used when multiplying numbers, it does not work with the sum! And we must not forget that this and the following properties apply only to powers with the same base.

4th property.

If the number in the denominator is raised to a negative power, then when subtracting, the degree of the denominator is taken in brackets to correctly replace the sign in further calculations.

The property only works when dividing, not when subtracting!

5th property.

6th property.

This property can also be applied to reverse side. A unit divided by a number to some degree is that number to a negative power.

7th property.

This property cannot be applied to sum and difference! When raising a sum or difference to a power, abbreviated multiplication formulas are used, not the properties of the power.

8th property.

9th property.

This property works for any fractional degree with a numerator equal to one, the formula will be the same, only the degree of the root will change depending on the denominator of the degree.

Also, this property is often used in reverse order. The root of any power of a number can be represented as that number to the power of one divided by the power of the root. This property is very useful in cases where the root of the number is not extracted.

10th property.

This property works not only with square root and second degree. If the degree of the root and the degree to which this root is raised are the same, then the answer will be a radical expression.

11th property.

You need to be able to see this property in time when solving it in order to save yourself from huge calculations.

12th property.

Each of these properties will meet you more than once in tasks, it can be given in its pure form, or it may require some transformations and the use of other formulas. Therefore, for the correct solution, it is not enough to know only the properties, you need to practice and connect the rest of mathematical knowledge.

Application of degrees and their properties

They are actively used in algebra and geometry. Degrees in mathematics have a separate, important place. With their help, exponential equations and inequalities are solved, as well as powers often complicate equations and examples related to other sections of mathematics. Exponents help to avoid large and long calculations, it is easier to reduce and calculate the exponents. But to work with large degrees, or with degrees big numbers, you need to know not only the properties of the degree, but also competently work with the bases, be able to decompose them in order to make your task easier. For convenience, you should also know the meaning of numbers raised to a power. This will reduce your time in solving by eliminating the need for long calculations.

The concept of degree plays a special role in logarithms. Since the logarithm, in essence, is the power of a number.

Abbreviated multiplication formulas are another example of the use of powers. They cannot use the properties of degrees, they are decomposed according to special rules, but every abbreviated multiplication formula invariably contains powers.

Degrees are also actively used in physics and computer science. All translations into the SI system are made using degrees, and in the future, when solving problems, the properties of the degree are applied. In computer science, powers of two are actively used, for the convenience of counting and simplifying the perception of numbers. Further calculations for conversions of units of measurement or calculations of problems, just as in physics, occur using the properties of the degree.

Degrees are also very useful in astronomy, where you can rarely find the use of the properties of a degree, but the degrees themselves are actively used to shorten the recording of various quantities and distances.

Degrees are also used in everyday life, when calculating areas, volumes, distances.

With the help of degrees, very large and very small values ​​\u200b\u200bare written in any field of science.

exponential equations and inequalities

Degree properties occupy a special place precisely in exponential equations and inequalities. These tasks are very common, as in school course as well as in exams. All of them are solved by applying the properties of the degree. The unknown is always in the degree itself, therefore, knowing all the properties, it will not be difficult to solve such an equation or inequality.

Obviously, numbers with powers can be added like other quantities , by adding them one by one with their signs.

So, the sum of a 3 and b 2 is a 3 + b 2 .
The sum of a 3 - b n and h 5 -d 4 is a 3 - b n + h 5 - d 4 .

Odds the same powers of the same variables can be added or subtracted.

So, the sum of 2a 2 and 3a 2 is 5a 2 .

It is also obvious that if we take two squares a, or three squares a, or five squares a.

But degrees various variables And various degrees identical variables, must be added by adding them to their signs.

So, the sum of a 2 and a 3 is the sum of a 2 + a 3 .

It is obvious that the square of a, and the cube of a, is neither twice the square of a, but twice the cube of a.

The sum of a 3 b n and 3a 5 b 6 is a 3 b n + 3a 5 b 6 .

Subtraction powers are carried out in the same way as addition, except that the signs of the subtrahend must be changed accordingly.

Or:
2a 4 - (-6a 4) = 8a 4
3h 2 b 6 - 4h 2 b 6 = -h 2 b 6
5(a - h) 6 - 2(a - h) 6 = 3(a - h) 6

Power multiplication

Numbers with powers can be multiplied like other quantities by writing them one after the other, with or without the multiplication sign between them.

So, the result of multiplying a 3 by b 2 is a 3 b 2 or aaabb.

Or:
x -3 ⋅ a m = a m x -3
3a 6 y 2 ⋅ (-2x) = -6a 6 xy 2
a 2 b 3 y 2 ⋅ a 3 b 2 y = a 2 b 3 y 2 a 3 b 2 y

The result in the last example can be ordered by adding the same variables.
The expression will take the form: a 5 b 5 y 3 .

By comparing several numbers (variables) with powers, we can see that if any two of them are multiplied, then the result is a number (variable) with a power equal to sum degrees of terms.

So, a 2 .a 3 = aa.aaa = aaaaa = a 5 .

Here 5 is the power of the result of the multiplication, equal to 2 + 3, the sum of the powers of the terms.

So, a n .a m = a m+n .

For a n , a is taken as a factor as many times as the power of n is;

And a m , is taken as a factor as many times as the degree m is equal to;

That's why, powers with the same bases can be multiplied by adding the exponents.

So, a 2 .a 6 = a 2+6 = a 8 . And x 3 .x 2 .x = x 3+2+1 = x 6 .

Or:
4a n ⋅ 2a n = 8a 2n
b 2 y 3 ⋅ b 4 y = b 6 y 4
(b + h - y) n ⋅ (b + h - y) = (b + h - y) n+1

Multiply (x 3 + x 2 y + xy 2 + y 3) ⋅ (x - y).
Answer: x 4 - y 4.
Multiply (x 3 + x - 5) ⋅ (2x 3 + x + 1).

This rule is also true for numbers whose exponents are - negative.

1. So, a -2 .a -3 = a -5 . This can be written as (1/aa).(1/aaa) = 1/aaaaa.

2. y-n .y-m = y-n-m .

3. a -n .a m = a m-n .

If a + b are multiplied by a - b, the result will be a 2 - b 2: that is

The result of multiplying the sum or difference of two numbers is equal to the sum or difference of their squares.

If the sum and difference of two numbers raised to square, the result will be equal to the sum or difference of these numbers in fourth degree.

So, (a - y).(a + y) = a 2 - y 2 .
(a 2 - y 2)⋅(a 2 + y 2) = a 4 - y 4 .
(a 4 - y 4)⋅(a 4 + y 4) = a 8 - y 8 .

Division of powers

Numbers with powers can be divided like other numbers by subtracting from the divisor, or by placing them in the form of a fraction.

So a 3 b 2 divided by b 2 is a 3 .

Or:
$\frac(9a^3y^4)(-3a^3) = -3y^4$
$\frac(a^2b + 3a^2)(a^2) = \frac(a^2(b+3))(a^2) = b + 3$
$\frac(d\cdot (a - h + y)^3)((a - h + y)^3) = d$

Writing a 5 divided by a 3 looks like $\frac(a^5)(a^3)$. But this is equal to a 2 . In a series of numbers
a +4 , a +3 , a +2 , a +1 , a 0 , a -1 , a -2 , a -3 , a -4 .
any number can be divided by another, and the exponent will be equal to difference indicators of divisible numbers.

When dividing powers with the same base, their exponents are subtracted..

So, y 3:y 2 = y 3-2 = y 1 . That is, $\frac(yyy)(yy) = y$.

And a n+1:a = a n+1-1 = a n . That is, $\frac(aa^n)(a) = a^n$.

Or:
y2m: ym = ym
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b + y) n-3

The rule is also valid for numbers with negative degree values.
The result of dividing a -5 by a -3 is a -2 .
Also, $\frac(1)(aaaaa) : \frac(1)(aaa) = \frac(1)(aaaaa).\frac(aaa)(1) = \frac(aaa)(aaaaa) = \frac (1)(aa)$.

h 2:h -1 = h 2+1 = h 3 or $h^2:\frac(1)(h) = h^2.\frac(h)(1) = h^3$

It is necessary to master the multiplication and division of powers very well, since such operations are very widely used in algebra.

Examples of solving examples with fractions containing numbers with powers

1. Reduce the exponents in $\frac(5a^4)(3a^2)$ Answer: $\frac(5a^2)(3)$.

2. Reduce the exponents in $\frac(6x^6)(3x^5)$. Answer: $\frac(2x)(1)$ or 2x.

3. Reduce the exponents a 2 / a 3 and a -3 / a -4 and bring to a common denominator.
a 2 .a -4 is a -2 first numerator.
a 3 .a -3 is a 0 = 1, the second numerator.
a 3 .a -4 is a -1 , the common numerator.
After simplification: a -2 /a -1 and 1/a -1 .

4. Reduce the exponents 2a 4 /5a 3 and 2 /a 4 and bring to a common denominator.
Answer: 2a 3 / 5a 7 and 5a 5 / 5a 7 or 2a 3 / 5a 2 and 5/5a 2.

5. Multiply (a 3 + b)/b 4 by (a - b)/3.

6. Multiply (a 5 + 1)/x 2 by (b 2 - 1)/(x + a).

7. Multiply b 4 /a -2 by h -3 /x and a n /y -3 .

8. Divide a 4 /y 3 by a 3 /y 2 . Answer: a/y.

9. Divide (h 3 - 1)/d 4 by (d n + 1)/h.