The degree is used to simplify the recording of the multiplication of the number of themselves. For example, instead of recording you can write 4 5 (\\ DisplayStyle 4 ^ (5)) (An explanation of this transition is given in the first section of this article). Degrees allow you to simplify writing long or complex expressions or equations; Also, the degrees are easily folded and subtracted, which leads to a simplification of expression or equation (for example, 4 2 * 4 3 \u003d 4 5 (\\ displaystyle 4 ^ (2) * 4 ^ (3) \u003d 4 ^ (5))).

Note: If you need to solve the indicative equation (in this equation, the unknown is in an indicator of the degree), read.

Steps

Solution of the simplest tasks with degrees

Multiply the foundation of the degree itself by the number of times equal to the indicator. If you need to solve the task with degrees manually, rewrite the degree in the form of a multiplication operation, where the foundation of the degree is multiplied by itself. For example, given a degree 3 4 (\\ DisplayStyle 3 ^ (4)). In this case, the base of degree 3 must be multiplied by itself 4 times: 3 * 3 * 3 * 3 (\\ DISPLAYSTYLE 3 * 3 * 3 * 3). Here are other examples:

To begin with, multiply the first two numbers. For example, 4 5 (\\ DisplayStyle 4 ^ (5)) = 4 * 4 * 4 * 4 * 4 (\\ DISPLAYSTYLE 4 * 4 * 4 * 4 * 4). Do not worry - the process of calculation is not so complicated as it seems at first glance. First, multiply the first two fours, and then replace them with the result. Like this:

• 4 5 \u003d 4 * 4 * 4 * 4 * 4 (\\ displaystyle 4 ^ (5) \u003d 4 * 4 * 4 * 4 * 4)
  • 4 * 4 \u003d 16 (\\ DISPLAYSTYLE 4 * 4 \u003d 16)

1. Multiply the result (in our example 16) to the next number. Each subsequent result will increase in proportion to. In our example, multiply 16 by 4. So:

   • 4 5 \u003d 16 * 4 * 4 * 4 (\\ DISPLAYSTYLE 4 ^ (5) \u003d 16 * 4 * 4 * 4)
     • 16 * 4 \u003d 64 (\\ displayStyle 16 * 4 \u003d 64)
   • 4 5 \u003d 64 * 4 * 4 (\\ displaystyle 4 ^ (5) \u003d 64 * 4 * 4)
     • 64 * 4 \u003d 256 (\\ DisplayStyle 64 * 4 \u003d 256)
   • 4 5 \u003d 256 * 4 (\\ displayStyle 4 ^ (5) \u003d 256 * 4)
     • 256 * 4 \u003d 1024 (\\ DisplayStyle 256 * 4 \u003d 1024)
   • Continue multiplying the result of multiplying the first two numbers to the next number until you receive the final answer. To do this, change the first two numbers, and then the result is multiplied by the next number in the sequence. This method is valid for any degree. In our example you should get: 4 5 \u003d 4 * 4 * 4 * 4 * 4 \u003d 1024 (\\ displaystyle 4 ^ (5) \u003d 4 * 4 * 4 * 4 * 4 \u003d 1024) .

2. Decide the following tasks. Check the answer using the calculator.

   • 8 2 (\\ DisplayStyle 8 ^ (2))
   • 3 4 (\\ DisplayStyle 3 ^ (4))
   • 10 7 (\\ DisplayStyle 10 ^ (7))

3. On the calculator, find the key indicated as "exp", or " x n (\\ displaystyle x ^ (n))", Or" ^ ". With this key, you will raise the number in the degree. Calculate the extent with a large indicator manually impossible (for example, degree 9 15 (\\ DisplayStyle 9 ^ (15))), But the calculator can easily cope with this task. In Windows 7, the standard calculator can be switched to engineering mode; To do this, click "View" -\u003e "Engineering". To switch to normal mode, click "View" -\u003e "Normal".

   • Check the response to the search engine (Google or Yandex). Using the "^" key on the computer keyboard, enter the search engine expression, which instantly displays the correct answer (and may offer similar expressions to study).

   Addition, subtraction, multiplication of degrees

   1. You can add and deduct degrees only if they have the same bases. If you need to add degrees with the same bases and indicators, you can replace the operation of the addition of multiplication. For example, the expression is given 4 5 + 4 5 (\\ displaystyle 4 ^ (5) + 4 ^ (5)). Remember that the degree 4 5 (\\ DisplayStyle 4 ^ (5)) can be represented as 1 * 4 5 (\\ DisplayStyle 1 * 4 ^ (5)); in this way, 4 5 + 4 5 \u003d 1 * 4 5 + 1 * 4 5 \u003d 2 * 4 5 (\\ DISPLAYSTYLE 4 ^ (5) + 4 ^ (5) \u003d 1 * 4 ^ (5) + 1 * 4 ^ (5) \u003d 2 * 4 ^ (5)) (where 1 +1 \u003d 2). That is, consider the number of similar degrees, and then multiply such a degree and this is the number. In our example, build 4 in the fifth degree, and then the resulting result multiply by 2. Remember that the addition operation can be replaced by multiplication operation, for example, 3 + 3 \u003d 2 * 3 (\\ displaystyle 3 + 3 \u003d 2 * 3). Here are other examples:

      • 3 2 + 3 2 \u003d 2 * 3 2 (\\ DISPLAYSTYLE 3 ^ (2) + 3 ^ (2) \u003d 2 * 3 ^ (2))
      • 4 5 + 4 5 + 4 5 \u003d 3 * 4 5 (\\ displaystyle 4 ^ (5) + 4 ^ (5) + 4 ^ (5) \u003d 3 * 4 ^ (5))
      • 4 5 - 4 5 + 2 \u003d 2 (\\ displaystyle 4 ^ (5) -4 ^ (5) + 2 \u003d 2)
      • 4 x 2 - 2 x 2 \u003d 2 x 2 (\\ displaystyle 4x ^ (2) -2x ^ (2) \u003d 2x ^ (2))

   2. When multiplying degrees with the same base, their indicators are folded (the base does not change). For example, the expression is given x 2 * x 5 (\\ displaystyle x ^ (2) * x ^ (5)). In this case, you just need to fold the indicators, leaving the basis unchanged. In this way, x 2 * x 5 \u003d x 7 (\\ displaystyle x ^ (2) * x ^ (5) \u003d x ^ (7)). Here is a visual explanation of this rule:

      When erecting a degree, the indicators are multiplied. For example, given a degree. Since the indicators of the degree are variable, then (x 2) 5 \u003d x 2 * 5 \u003d x 10 (\\ displaystyle (x ^ (2)) ^ (5) \u003d x ^ (2 * 5) \u003d x ^ (10)). The meaning of this rule is that you multiply the degree (x 2) (\\ displaystyle (x ^ (2))) Self for himself five times. Like this:

      • (x 2) 5 (\\ displaystyle (x ^ (2)) ^ (5))
      • (x 2) 5 \u003d x 2 * x 2 * x 2 * x 2 * x 2 (\\ displaystyle (x ^ (2)) ^ (5) \u003d x ^ (2) * x ^ (2) * x ^ ( 2) * x ^ (2) * x ^ (2))
      • Since the base is the same, the indicators of the degree simply add up: (x 2) 5 \u003d x 2 * x 2 * x 2 * x 2 * x 2 \u003d x 10 (\\ displaystyle (x ^ (2)) ^ (5) \u003d x ^ (2) * x ^ (2) * x ^ (2) * x ^ (2) * x ^ (2) \u003d x ^ (10))

   3. The degree with a negative indicator should be converted to a fraction (to the inverse degree). Not trouble, if you do not know what the return degree is. If you are given a degree with a negative indicator, for example, 3 - 2 (\\ displaystyle 3 ^ (- 2)), Write this degree into the trousers denominator (in the Numerator, place 1), and make the indicator positive. In our example: 1 3 2 (\\ displayStyle (\\ FRAC (1) (3 ^ (2)))). Here are other examples:

      When dividing degrees with the same base, their indicators are deducted (the basis does not change). The division operation is the opposite of multiplication operation. For example, the expression is given 4 4 4 2 (\\ displayStyle (\\ FRAC (4 ^ (4)) (4 ^ (2)))). Delete the degree in the denominator, from the indicator of the degree facing the numerator (do not change the base). In this way, 4 4 4 2 \u003d 4 4 - 2 \u003d 4 2 (\\ displayStyle (\\ FRAC (4 ^ (4)) (4 ^ (2))) \u003d 4 ^ (4-2) \u003d 4 ^ (2)) = 16 .

      • The degree facing the denominator can be written in this form: 1 4 2 (\\ DisplayStyle (\\ FRAC (1) (4 ^ (2)))) = 4 - 2 (\\ displaystyle 4 ^ (- 2)). Remember that the fraction is a number (degree, expression) with a negative indicator of the degree.

   4. Below are some expressions that will help you learn to solve tasks with degrees. These expressions cover the material set out in this section. In order to see the answer, simply highlight the empty space after the equality sign.

   Solving tasks with fractional indicators

   The degree with a fractional indicator (for example,) is converted to the operation of the root extraction. In our example: x 1 2 (\\ displaystyle x ^ (\\ FRAC (1) (2))) = X (\\ DisplayStyle (\\ SQRT (X))). It does not matter what number is in the denominator of the fractional indicator. For example, x 1 4 (\\ displayStyle X ^ (\\ FRAC (1) (4))) - this is the root of the fourth degree from "x", that is x 4 (\\ DisplayStyle (\\ SQRT [(4)] (x))) .

   1. If the indicator of the degree is irregular fraction, then such a degree can be decomposed for two degrees to simplify the solution of the problem. There is nothing complicated in this - just remember the rule of multiplication by degrees. For example, given a degree. Turn such a degree to the root, the degree of which will be equal to the protruser of the fractional indicator, and then take this root to the degree equal to the numerator of the fractional indicator. To do it, remember that 5 3 (\\ DisplayStyle (\\ FRAC (5) (3))) = (1 3) * 5 (\\ displayStyle ((\\ FRAC (1) (3))) * 5). In our example:

      • x 5 3 (\\ displaystyle x ^ (\\ FRAC (5) (3)))
      • x 1 3 \u003d x 3 (\\ displaystyle x ^ (\\ FRAC (1) (3)) \u003d (\\ sqrt [(3)] (x)))
      • x 5 3 \u003d x 5 * x 1 3 (\\ displaystyle x ^ (\\ FRAC (5) (3)) \u003d x ^ (5) * x ^ (\\ FRAC (1) (3))) = (x 3) 5 (\\ displaystyle ((\\ sqrt [(3)] (x))) ^ (5))
   2. On some calculators there is a button to calculate degrees (you first need to enter the base, then press the button, and then enter the indicator). It is denoted as ^ or x ^ y.
   3. Remember that any number in the first degree equally to itself, for example, 4 1 \u003d 4. (\\ displaystyle 4 ^ (1) \u003d 4.) Moreover, any number multiplied or divided by one is equal to itself, for example, 5 * 1 \u003d 5 (\\ DisplayStyle 5 * 1 \u003d 5) and 5/1 \u003d 5 (\\ DisplayStyle 5/1 \u003d 5).
   4. Know that degrees 0 0 does not exist (such a degree has no solution). When you try to solve such a degree on the calculator or on the computer you will get an error. But remember that any number in zero degree is 1, for example, 4 0 \u003d 1. (\\ displaystyle 4 ^ (0) \u003d 1.)
   5. In the highest mathematics, which operates with imaginary numbers: e a i x \u003d c o s a x + i s i n a x (\\ displaystyle e ^ (a) ix \u003d cosax + isinax)where i \u003d (- 1) (\\ displaystyle i \u003d (\\ sqrt (()) - 1)); e - constant, approximately equal to 2.7; A - arbitrary constant. The proof of this equality can be found in any textbook on higher mathematics.

   Warnings

   • With an increase in the indicator of the degree, its value increases much. Therefore, if the answer seems to you wrong, in fact he may be faithful. You can check it out by building a schedule of any indicative function, for example, 2 x.

Consider the topic of transforming expressions with degrees, but first let's stop at a number of transformations that can be carried out with any expressions, including with power. We will learn to reveal brackets, bring similar terms, to work with the basis and indicator of the degree, use the properties of degrees.

Yandex.rtb R-A-339285-1

What are powerful expressions?

IN school course Few uses of the phrase "powerful expressions", but this term is constantly meeting in collections to prepare for the exam. In most cases, phrases are indicated by expressions that contain in their degree records. This is we reflect in our definition.

Definition 1.

Power expression - This is an expression that contains degrees.

Let us give a few examples of power expressions, starting with the degree with a natural indicator and ending with the real indicator.

The most simple power expressions can be considered the degree of the number with the natural indicator: 3 2, 7 5 + 1, (2 + 1) 5, (- 0, 1) 4, 2 2 3 3, 3 · A 2 - A + A 2, x 3 - 1, (a 2) 3. As well as degrees with zero indicator: 5 0, (A + 1) 0, 3 + 5 2 - 3, 2 0. And degrees with whole negative degrees: (0, 5) 2 + (0, 5) - 2 2.

Easily more difficult to work with a degree having rational and irrational indicators: 264 1 4 - 3 · 3 · 3 1 2, 2 3, 5 · 2 - 2 2 - 1, 5, 1 A 1 4 · A 1 2 - 2 · A - 1 6 · b 1 2, x π · x 1 - π, 2 3 3 + 5.

As an indicator, a variable 3 x - 54 - 7 · 3 x - 58 or logarithm can be x 2 · L g x - 5 · x L g x.

With the question of what power expressions are, we figured out. Now we will deal with their conversion.

The main types of transformations of power expressions

First of all, we will consider the basic identity transformations of expressions that can be performed with power expressions.

Example 1.

Calculate the value of the power expression 2 3 · (4 2 - 12).

Decision

All transformations we will be carried out in compliance with the procedure for performing actions. In this case, we begin with the implementation of actions in brackets: replace the degree of digital value and calculate the difference of two numbers. Have 2 3 · (4 2 - 12) \u003d 2 3 · (16 - 12) \u003d 2 3 · 4.

We still have to replace the degree 2 3 His meaning 8 and calculate the work 8 · 4 \u003d 32. Here is our answer.

Answer: 2 3 · (4 2 - 12) \u003d 32.

Example 2.

Simplify expression with degrees 3 · a 4 · b - 7 - 1 + 2 · a 4 · b - 7.

Decision

An expression given to us in terms of the task contains similar terms that we can lead: 3 · a 4 · b - 7 - 1 + 2 · a 4 · b - 7 \u003d 5 · a 4 · b - 7 - 1.

Answer: 3 · a 4 · b - 7 - 1 + 2 · a 4 · b - 7 \u003d 5 · a 4 · b - 7 - 1.

Example 3.

Prepare an expression with degrees 9 - b 3 · π - 1 2 as a piece.

Decision

Imagine number 9 as a degree 3 2 and apply the formula of abbreviated multiplication:

9 - B 3 · π - 1 2 \u003d 3 2 - B 3 · π - 1 2 \u003d \u003d 3 - B 3 · π - 1 3 + B 3 · π - 1

Answer: 9 - B 3 · π - 1 2 \u003d 3 - B 3 · π - 1 3 + B 3 · π - 1.

And now we turn to the disaster identical transformationswhich can be used precisely in relation to power expressions.

Work with the basis and indicator of the degree

The degree in the base or indicator may also have numbers, variables, and some expressions. For example, (2 + 0, 3 · 7) 5 - 3, 7 and . Work with such entries is difficult. It is much easier to replace the expression at the base of the degree or expression in the indicator identically equal to an expression.

Degree and indicator transformations are carried out according to the rules known to us separately from each other. The most important thing is that as a result of the transformation, an expression is identical to the initial one.

The purpose of the transformations is to simplify the initial expression or get the solution to the problem. For example, in the example, which we led above, (2 + 0, 3 · 7) 5 - 3, 7, you can perform actions to transition to degree 4 , 1 1 , 3 . Open brackets, we can lead similar terms at the bottom (A · (A + 1) - a 2) 2 · (x + 1) and get a powerful expression of a simpler type a 2 · (x + 1).

Use the properties of degrees

The properties of degrees recorded in the form of equalities are one of the main tools for transforming expressions with degrees. Here are the main of them, given that A. and B. - these are any positive numbers, and R. and S. - arbitrary valid numbers:

Definition 2.

• a r · a s \u003d a r + s;
• a r: a s \u003d a r - s;
• (a · b) r \u003d a r · b r;
• (A: B) R \u003d A R: B R;
• (A R) S \u003d A R · s.

In cases where we are dealing with natural, integer, positive indicators of the degree, the limitations on the number A and B may be much less strict. So, for example, if we consider equality a m · a n \u003d a m + nwhere M. and N. - natural numbers, it will be true for any values \u200b\u200bof A, both positive and negative, as well as for a \u003d 0..

It is possible to apply the properties of degrees without restrictions in cases where the bases of degrees are positive or contain variables, the area of \u200b\u200bpermissible values \u200b\u200bof which is such that only positive values \u200b\u200bare taken on it. In fact, within school program In mathematics, the student's task is to choose a suitable property and its correct application.

When preparing for admission to universities, tasks may occur in which the inaccient use of properties will lead to a narrowing of OTZ and other difficulties with the solution. In this section, we will analyze only two such cases. More information on the issue can be found in the topic "Transformation of expressions using the properties of degrees".

Example 4.

Imagine an expression A 2, 5 · (A 2) - 3: A - 5, 5 in the form of a degree A..

Decision

To begin with, we use the exercise property and we transform the second factor on it. (A 2) - 3 . Then use the properties of multiplication and division of degrees with the same base:

a 2, 5 · a - 6: a - 5, 5 \u003d a 2, 5 - 6: a - 5, 5 \u003d a - 3, 5: a - 5, 5 \u003d a - 3, 5 - (- 5, 5) \u003d a 2.

Answer: A 2, 5 · (A 2) - 3: A - 5, 5 \u003d A 2.

The transformation of power expressions according to the property of degrees can be made both from left to right and in the opposite direction.

Example 5.

Find the value of the power expression 3 1 3 · 7 1 3 · 21 2 3.

Decision

If we apply equality (A · b) r \u003d a r · b r, right to left, then we get a product of the form 3 · 7 1 3 · 21 2 3 and further 21 1 3 · 21 2 3. Moving the indicators when multiplying degrees with the same bases: 21 1 3 · 21 2 3 \u003d 21 1 3 + 2 3 \u003d 21 1 \u003d 21.

There is another way to carry out conversion:

3 1 3 · 7 1 3 · 21 2 3 \u003d 3 1 3 · 7 1 3 · (3 · 7) 2 3 \u003d 3 1 3 · 7 1 3 · 3 2 3 · 7 2 3 \u003d 3 1 3 · 3 2 3 · 7 1 3 · 7 2 3 \u003d 3 1 3 + 2 3 · 7 1 3 + 2 3 \u003d 3 1 · 7 1 \u003d 21

Answer: 3 1 3 · 7 1 3 · 21 2 3 \u003d 3 1 · 7 1 \u003d 21

Example 6.

Power expression is given A 1, 5 - A 0, 5 - 6Enter a new variable T \u003d A 0, 5.

Decision

Imagine a degree A 1, 5 as a 0, 5 · 3 . Use the degree property to the degree (a r) s \u003d a r · s On the right left and obtain (a 0, 5) 3: a 1, 5 - a 0, 5 - 6 \u003d (a 0, 5) 3 - a 0, 5 - 6. In the resulting expression, you can easily enter a new variable. T \u003d A 0, 5: Receive T 3 - T - 6.

Answer: T 3 - T - 6.

Transformation of fractions containing degrees

We usually deal with two variants of power expressions with fractions: the expression is a fraction with a degree or contains such a fraction. These expressions apply all major transformations of fractions without restrictions. They can be reduced, lead to a new denominator, work separately with a numerator and denominator. We illustrate this by examples.

Example 7.

Simplify the power expression 3 · 5 2 3 · 5 1 3 - 5 - 2 3 1 + 2 · x 2 - 3 - 3 · x 2.

Decision

We are dealing with a fraction, so we carry out transformations in the numerator, and in the denominator:

3 · 5 2 3 · 5 1 3 - 5 - 2 3 1 + 2 · x 2 - 3 - 3 · x 2 \u003d 3 · 5 2 3 · 5 1 3 - 3 · 5 2 3 · 5 - 2 3 - 2 - x 2 \u003d \u003d 3 · 5 2 3 + 1 3 - 3 · 5 2 3 + - 2 3 - 2 - x 2 \u003d 3 · 5 1 - 3 · 5 0 - 2 - x 2

Position minus before the fraction in order to change the sign of the denominator: 12 - 2 - x 2 \u003d - 12 2 + x 2

Answer: 3 · 5 2 3 · 5 1 3 - 5 - 2 3 1 + 2 · x 2 - 3 - 3 · x 2 \u003d - 12 2 + x 2

The fractions containing degrees are given to the new denominator in exactly as well as rational fractions. To do this, you need to find an additional multiplier and multiply the numerator and denominator of the fraction. It is necessary to select an additional factor in such a way that it does not apply to zero under any values \u200b\u200bof the variables from the odd variables for the initial expression.

Example 8.

Give fractions to a new denominator: a) A + 1 A 0, 7 to the denominator A., b) 1 x 2 3 - 2 · x 1 3 · y 1 6 + 4 · y 1 3 to denominator X + 8 · Y 1 2.

Decision

a) We will select a multiplier who will allow us to bring to a new denominator. a 0, 7 · a 0, 3 \u003d a 0, 7 + 0, 3 \u003d a,therefore, as an additional multiplier we will take A 0, 3. The area of \u200b\u200bpermissible values \u200b\u200bof the variable A includes many of all positive valid numbers. In this area A 0, 3 Not accessed to zero.

Perform multiplication of the numerator and denominator of the fraction on A 0, 3:

a + 1 a 0, 7 \u003d a + 1 · a 0, 3 a 0, 7 · a 0, 3 \u003d a + 1 · a 0, 3 a

b) pay attention to the denominator:

x 2 3 - 2 · x 1 3 · y 1 6 + 4 · y 1 3 \u003d x 1 3 2 - x 1 3 · 2 · y 1 6 + 2 · y 1 6 2

Multiply this expression on x 1 3 + 2 · Y 1 6, we obtain the sum of cubes x 1 3 and 2 · y 1 6, i.e. X + 8 · Y 1 2. This is our new denominator to which we need to bring the original fraction.

So we found an additional multiplier X 1 3 + 2 · Y 1 6. On the area of \u200b\u200bpermissible values \u200b\u200bof variables X. and y. The expression x 1 3 + 2 · Y 1 6 does not turn to zero, so we can multiply the numerator and denominator of the fraction:
1 x 2 3 - 2 · x 1 3 · y 1 6 + 4 · y 1 3 \u003d x 1 3 + 2 · y 1 6 x 1 3 + 2 · y 1 6 x 2 3 - 2 · x 1 3 · y 1 6 + 4 · y 1 3 \u003d x 1 3 + 2 · y 1 6 x 1 3 3 + 2 · y 1 6 3 \u003d x 1 3 + 2 · y 1 6 x + 8 · y 1 2

Answer: a) a + 1 a 0, 7 \u003d a + 1 · a 0, 3 a, b) 1 x 2 3 - 2 · x 1 3 · y 1 6 + 4 · y 1 3 \u003d x 1 3 + 2 · y 1 6 x + 8 · y 1 2.

Example 9.

Reduce the fraction: a) 30 · x 3 · (x 0, 5 + 1) · x + 2 · x 1 1 3 - 5 3 45 · x 0, 5 + 1 2 · x + 2 · x 1 1 3 - 5 3, b) a 1 4 - b 1 4 A 1 2 - B 1 2.

Decision

a) We use the largest common denominator (node) to which the numerator and denominator can be reduced. For numbers 30 and 45, this is 15. We can also reduce on x 0, 5 + 1 and on x + 2 · x 1 1 3 - 5 3.

We get:

30 · x 3 · (x 0, 5 + 1) · x + 2 · x 1 1 3 - 5 3 45 · x 0, 5 + 1 2 · x + 2 · x 1 1 3 - 5 3 \u003d 2 · x 3 3 · (x 0, 5 + 1)

b) Here the presence of the same multipliers is not obvious. You will have to perform some conversions in order to obtain the same multipliers in a numerator and denominator. To do this, lay an denominator using the square difference formula:

a 1 4 - B 1 4 A 1 2 - B 1 2 \u003d A 1 4 - B 1 4 A 1 4 2 - B 1 2 2 \u003d A 1 4 - B 1 4 A 1 4 + B 1 4 · A 1 4 - B 1 4 \u003d 1 A 1 4 + B 1 4

Answer:a) 30 · x 3 · (x 0, 5 + 1) · x + 2 · x 1 1 3 - 5 3 45 · x 0, 5 + 1 2 · x + 2 · x 1 1 3 - 5 3 \u003d 2 · X 3 3 · (x 0, 5 + 1), b) a 1 4 - b 1 4 A 1 2 - b 1 2 \u003d 1 A 1 4 + B 1 4.

The essential actions with fractions include bringing to a new denominator and cutting fractions. Both actions are performed in compliance with a number of rules. When adding and subtracting fractions first, the fractions are given to common denominator, after which actions are carried out (addition or subtraction) with numerators. The denominator remains the same. The result of our actions is a new fraction, the numerator of which is the product of numerators, and the denominator is a product of denominators.

Example 10.

Perform actions x 1 2 + 1 x 1 2 - 1 - x 1 2 - 1 x 1 2 + 1 · 1 x 1 2.

Decision

Let's start with the subtraction of fractions that are located in brackets. We give them to the general denominator:

x 1 2 - 1 · x 1 2 + 1

Subscribe numbers:

x 1 2 + 1 x 1 2 - 1 - x 1 2 - 1 x 1 2 + 1 · 1 x 1 2 \u003d x 1 2 + 1 · x 1 2 + 1 x 1 2 - 1 · x 1 2 + 1 - x 1 2 - 1 · x 1 2 - 1 x 1 2 + 1 · x 1 2 - 1 · 1 x 1 2 \u003d x 1 2 + 1 2 - x 1 2 - 1 2 x 1 2 - 1 · x 1 2 + 1 · 1 x 1 2 \u003d x 1 2 2 + 2 · x 1 2 + 1 - x 1 2 2 - 2 · x 1 2 + 1 x 1 2 - 1 · x 1 2 + 1 · 1 x 1 2 \u003d \u003d 4 · x 1 2 x 1 2 - 1 · x 1 2 + 1 · 1 x 1 2

Now we multiply the fractions:

4 · x 1 2 x 1 2 - 1 · x 1 2 + 1 · 1 x 1 2 \u003d 4 · x 1 2 x 1 2 - 1 · x 1 2 + 1 · x 1 2

We will reduce to the degree x 1 2., we obtain 4 x 1 2 - 1 · x 1 2 + 1.

Additionally, it is possible to simplify the power expression in the denominator, using the square difference formula: Squares: 4 x 1 2 - 1 · x 1 2 + 1 \u003d 4 x 1 2 2 - 1 2 \u003d 4 x - 1.

Answer: x 1 2 + 1 x 1 2 - 1 - x 1 2 - 1 x 1 2 + 1 · 1 x 1 2 \u003d 4 x - 1

Example 11.

Simplify the power expression x 3 4 · x 2, 7 + 1 2 x - 5 8 · x 2, 7 + 1 3.
Decision

We can reduce the fraction on (x 2, 7 + 1) 2. We obtain the fraction x 3 4 x - 5 8 · x 2, 7 + 1.

We continue to transform the degrees of X 3 4 x - 5 8 · 1 x 2, 7 + 1. Now you can use the defenses of degrees with the same bases: x 3 4 x - 5 8 · 1 x 2, 7 + 1 \u003d x 3 4 - - 5 8 · 1 x 2, 7 + 1 \u003d x 1 1 8 · 1 x 2 , 7 + 1.

Go from the last work to the fraction x 1 3 8 x 2, 7 + 1.

Answer: x 3 4 · x 2, 7 + 1 2 x - 5 8 · x 2, 7 + 1 3 \u003d x 1 3 8 x 2, 7 + 1.

Multiplers with negative indicators in most cases are more convenient to transfer from the numerator to the denominator and back, changing the indicator sign. This action allows you to simplify the further solution. Let us give an example: a power expression (X + 1) - 0, 2 3 · X - 1 can be replaced by x 3 · (x + 1) 0, 2.

Transformation of expressions with roots and degrees

There are significant expressions in the tasks that contain not only degrees with fractional indicators, but also roots. Such expressions are desirable to bring only to roots or only to degrees. The transition to degrees is preferable, since they are easier to work with them. Such a transition is particularly preferable when the OTZ variables for the original expression makes it possible to replace the roots by degrees without the need to turn to the module or split OTZ into several gaps.

Example 12.

Prepare the expression x 1 9 · x · x 3 6 as a degree.

Decision

Area of \u200b\u200bpermissible variable values X. Determined by two inequalities x ≥ 0. and x · x 3 ≥ 0, which set many [ 0 , + ∞) .

On this set we have the right to move from the roots to the degrees:

x 1 9 · x · x 3 6 \u003d x 1 9 · x · x 1 3 1 6

Using the properties of degrees, simplifies the resulting power expression.

x 1 9 · x · x 1 3 1 6 \u003d x 1 9 · x 1 6 · x 1 3 1 6 \u003d x 1 9 · x 1 6 · x 1 · 1 3 · 6 \u003d x 1 9 · x 1 6 · X 1 18 \u003d x 1 9 + 1 6 + 1 18 \u003d x 1 3

Answer: x 1 9 · x · x 3 6 \u003d x 1 3.

Transformation of degrees with variables in the indicator

The conversion data simply simply produce if competently use the degree properties. For example, 5 2 · x + 1 - 3 · 5 x · 7 x - 14 · 7 2 · x - 1 \u003d 0.

We can replace the degree in the indicators of which there is a sum of some variable and the number. In the left side, this can be done with the first and last term of the left part of the expression:

5 2 · x · 5 1 - 3 · 5 x · 7 x - 14 · 7 2 · x · 7 - 1 \u003d 0, 5 · 5 2 · x - 3 · 5 x · 7 x - 2 · 7 2 · x \u003d 0.

Now share both parts of equality on 7 2 · X. This expression on the OTZ variable X receives only positive values:

5 · 5 - 3 · 5 x · 7 x - 2 · 7 2 · x 7 2 · x \u003d 0 7 2 · x, 5 · 5 2 · x 7 2 · x - 3 · 5 x · 7 x 7 2 · x - 2 · 7 2 · x 7 2 · x \u003d 0, 5 · 5 2 · x 7 2 · x - 3 · 5 x · 7 x 7 x · 7 x - 2 · 7 2 · x 7 2 · x \u003d 0.

We will reduce the fractions with degrees, we get: 5 · 5 2 · x 7 2 · x - 3 · 5 x 7 x - 2 \u003d 0.

Finally, the ratio of degrees with the same indicators is replaced by degrees of relations, which leads to an equation 5 · 5 7 2 · x - 3 · 5 7 x - 2 \u003d 0, which is equivalent to 5 · 5 7 x 2 - 3 · 5 7 x - 2 \u003d 0.

We introduce a new variable T \u003d 5 7 x, which reduces the solution of the initial indicative equation to the solution of the square equation 5 · T 2 - 3 · T - 2 \u003d 0.

Transformation of expressions with degrees and logarithms

Expressions containing the degree and logarithm recording are also found in tasks. An example of such expressions can be: 1 4 1 - 5 · LOG 2 3 or LOG 3 27 9 + 5 (1 - log 3 5) · Log 5 3. The transformation of such expressions is carried out using the above approaches and the properties of the logarithms, which we dismantled in detail in the topic "Transformation of logarithmic expressions".

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Formulas degrees Used in the process of abbreviation and simplify complex expressions, in solving equations and inequalities.

Number c. is an n.Little degree a. when:

Operations with degrees.

1. Multiplying the degree with the same basis, their indicators fold:

a M.· A n \u003d a m + n.

2. In dividing degrees with the same basis, their indicators are deducted:

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

(ABC ...) n \u003d a n · b n · C n ...

4. The degree of fraction is equal to the ratio of degrees of the divide and divider:

(A / B) n \u003d a n / b n.

5. Earring the degree to the degree, the indicators of degrees are prolonged:

(a m) n \u003d a m n.

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

for example. (2 · 3 · 5/15) ² \u003d 2² · 3² · 5² / 15 ² \u003d 900/225 \u003d 4.

Root operations.

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

2. root from the relationship equal to relation divide and divider roots:

3. When the root is erected, it is fairly built into this degree.

4. If you increase the degree of root in n. once and at the same time build in n.The degree of the feed number, the value of the root will not change:

5. If you reduce the root degree in n. once and at the same time extract the root n.degree from an undercurned number, the value of the root will not change:

Degree with a negative indicator.The degree of a certain number with an indisputable (whole) indicator is determined as a unit divided by the degree of the same number with an indicator equal to the absolute value of the non-positive indicator:

Formula a M.: a n \u003d a m - n can be used not only at m.> n. but also m.< n..

for example. a. 4: A 7 \u003d A 4 \u200b\u200b- 7 \u003d A -3.

To formula a M.: a n \u003d a m - n became fair as m \u003d N.The presence of a zero degree is needed.

The degree with the zero indicator.The degree of any number that is not equal to zero, with the zero indicator equals one.

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

Degree with fractional indicator.To build a valid number but in degree m / N., it is necessary to extract the root n.degree from m.degree of this number but.

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First, let's remember the basic formulas of the degrees and their properties.

The work of the number a. The itself occurs n times, this expression we can write down as a a A ... a \u003d a n

1. A 0 \u003d 1 (A ≠ 0)

3. a n a m \u003d a n + m

4. (a n) m \u003d a nm

5. A N B n \u003d (AB) N

7. A N / A M \u003d A N - M

Power or indicative equations - These are equations in which variables are in degrees (or indicators), and the basis is the number.

Examples of indicative equations:

In this example, the number 6 is the basis it always stands downstairs, and the variable x. degree or indicator.

Let us give more examples of the indicative equations.
2 x * 5 \u003d 10
16 x - 4 x - 6 \u003d 0

Now we will analyze how the demonstration equations are solved?

Take a simple equation:

2 x \u003d 2 3

This example can be solved even in the mind. It can be seen that x \u003d 3. After all, so that the left and right part should be equal to the number 3 instead of x.
Now let's see how it is necessary to issue this decision:

2 x \u003d 2 3
x \u003d 3.

In order to solve such an equation, we removed same grounds (i.e. two) and recorded what remains, it is degrees. Received the desired answer.

Now summarize our decision.

Algorithm for solving an indicative equation:
1. Need to check the same Lee foundations at the equation on the right and left. If the bases are not the same as looking for options for solving this example.
2. After the foundations become the same, equal degrees and solve the resulting new equation.

Now rewrite a few examples:

Let's start with a simple.

The bases in the left and right part are equal to Number 2, which means we can reject and equate their degrees.

x + 2 \u003d 4 It turned out the simplest equation.
x \u003d 4 - 2
x \u003d 2.
Answer: x \u003d 2

In the following example, it can be seen that the bases are different. It is 3 and 9.

3 3x - 9 x + 8 \u003d 0

To begin with, we transfer the nine to the right side, we get:

Now you need to make the same foundation. We know that 9 \u003d 3 2. We use the degree formula (a n) m \u003d a nm.

3 3x \u003d (3 2) x + 8

We obtain 9 x + 8 \u003d (3 2) x + 8 \u003d 3 2x + 16

3 3x \u003d 3 2x + 16 Now it is clear that in the left and right side The bases are the same and equal to the troika, then we can discard them and equate degrees.

3x \u003d 2x + 16 Received the simplest equation
3x - 2x \u003d 16
x \u003d 16.
Answer: X \u003d 16.

We look at the following example:

2 2x + 4 - 10 4 x \u003d 2 4

First, we look at the base, the foundations are different two and four. And we need to be the same. We convert the four by the formula (a n) m \u003d a nm.

4 x \u003d (2 2) x \u003d 2 2x

And also use one formula a n a m \u003d a n + m:

2 2x + 4 \u003d 2 2x 2 4

Add to equation:

2 2x 2 4 - 10 2 2x \u003d 24

We led an example to the same reasons. But we interfere with other numbers 10 and 24. What to do with them? If you can see that it is clear that we have 2 2 2, that's the answer - 2 2, we can take out the brackets:

2 2x (2 4 - 10) \u003d 24

We calculate the expression in brackets:

2 4 — 10 = 16 — 10 = 6

All equation Delim to 6:

Imagine 4 \u003d 2 2:

2 2x \u003d 2 2 bases are the same, throwing out them and equate degrees.
2x \u003d 2 It turned out the simplest equation. We divide it on 2
x \u003d 1.
Answer: x \u003d 1.

Resolving equation:

9 x - 12 * 3 x + 27 \u003d 0

We transform:
9 x \u003d (3 2) x \u003d 3 2x

We get the equation:
3 2x - 12 3 x +27 \u003d 0

The foundations we have the same are equal to three. In this example, it can be seen that the first three degree twice (2x) is greater than that of the second (simply x). In this case, you can solve replacement method. The number with the smallest degree replace:

Then 3 2x \u003d (3 x) 2 \u003d T 2

We replace in equation all degrees with cavities on T:

t 2 - 12T + 27 \u003d 0
Receive quadratic equation. We decide through the discriminant, we get:
D \u003d 144-108 \u003d 36
T 1 \u003d 9
T 2 \u003d 3

Return to the variable x..

Take T 1:
T 1 \u003d 9 \u003d 3 x

That is,

3 x \u003d 9
3 x \u003d 3 2
x 1 \u003d 2

One root found. We are looking for the second, from T 2:
T 2 \u003d 3 \u003d 3 x
3 x \u003d 3 1
x 2 \u003d 1
Answer: x 1 \u003d 2; x 2 \u003d 1.

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First level

The degree and properties. Exhaustive guide (2019)

Why are you needed? Where will they come to you? Why do you need to spend time on their study?

To find out all about the degrees, what they need for what they need how to use their knowledge in everyday life read this article.

And, of course, the knowledge of degrees will bring you closer to successful hand over fire or the exam and to enter the university of your dreams.

Let "S GO ... (drove!)

Important remark! If instead of formulas you see abracadabra, clean the cache. To do this, click Ctrl + F5 (on Windows) or CMD + R (on Mac).

FIRST LEVEL

The exercise is the same mathematical operation as addition, subtraction, multiplication or division.

Now I will explain all the human language on very simple examples. Pay attention. Examples of elementary, but explaining important things.

Let's start with addition.

There is nothing to explain here. You all know everything: we are eight people. Everyone has two bottles of cola. How much is the cola? Right - 16 bottles.

Now multiplication.

The same example with a cola can be recorded differently :. Mathematics - People cunning and lazy. They first notice some patterns, and then invent the way how to "count" them faster. In our case, they noticed that each of the eight people had the same number of cola bottles and came up with a reception called multiplication. Agree, it is considered easier and faster than.

So, to read faster, easier and without mistakes, you just need to remember table multiplication. Of course, you can do everything more slowly, harder and mistakes! But…

Here is the multiplication table. Repeat.

And the other, more beautiful:

And what other tricks came up with lazy mathematicians? Right - erection.

Erection

If you need to multiply the number for yourself five times, then mathematics say that you need to build this number in the fifth degree. For example, . Mathematics remember that two in the fifth degree is. And they solve such tasks in the mind - faster, easier and without errors.

For this you need only remember what is highlighted in color in the table of degrees of numbers. Believe it, it will greatly facilitate your life.

By the way, why the second degree is called square numbers, and the third - cuba? What does it mean? Very good question. Now there will be to you and squares, and Cuba.

Example from life number 1

Let's start with a square or from a second degree of number.

Imagine a square pool of meter size on a meter. The pool is on your dacha. Heat and really want to swim. But ... Pool without the bottom! You need to store the bottom of the pool tiles. How much do you need tiles? In order to determine this, you need to find out the area of \u200b\u200bthe bottom of the pool.

You can simply calculate, with a finger, that the bottom of the pool consists of a meter cubes per meter. If you have a meter tile for meter, you will need to pieces. It's easy ... But where did you see such a tile? The tile is more likely to see for see and then "finger to count" torture. Then you have to multiply. So, on one side of the bottom of the pool, we fit tiles (pieces) and on the other too tiles. Multiplying on, you will get tiles ().

Did you notice that in order to determine the area of \u200b\u200bthe bottom of the pool, did we multiply the same number by yourself? What does it mean? This is multiplied by the same number, we can take advantage of the "erection of the extermination". (Of course, when you have only two numbers, multiply them or raise them into the degree. But if you have a lot of them, it is much easier to raise them in terms of calculations, too much less. For the exam, it is very important).
So thirty to the second degree will (). Or we can say that thirty in the square will be. In other words, the second degree of number can always be represented as a square. And on the contrary, if you see a square - it is always the second degree of some number. Square is the image of a second degree number.

Example from life number 2

Here is the task, count how many squares on a chessboard with a square of the number ... on one side of the cells and on the other too. To calculate their quantity, you need to multiply eight or ... If you note that the chessboard is a square of the side, then you can build eight per square. It turns out cells. () So?

Example from life number 3

Now a cube or the third degree of number. The same pool. But now you need to know how much water will have to fill in this pool. You need to count the volume. (Volume and liquid, by the way, are measured in cubic meters. Suddenly, right?) Draw a pool: bottom of the meter size and a depth of meter and try to count how much cubes meter size per meter will enter your pool.

Right show your finger and count! Once, two, three, four ... twenty two, twenty three ... how much did it happen? Did not come down? Difficult to count your finger? So that! Take an example from mathematicians. They are lazy, therefore noticed that to calculate the volume of the pool, it is necessary to multiply each other in length, width and height. In our case, the volume of the pool will be equal to cubes ... it is easier for the truth?

And now imagine, as far as Mathematics are lazy and cunning, if they are simplified. Brought all to one action. They noticed that the length, width and height is equal to and that the same number varnims itself on itself ... And what does this mean? This means that you can take advantage of the degree. So, what did you think with your finger, they do in one action: three in Cuba is equal. This is written so :.

It remains only remember Table degrees. If you are, of course, the same lazy and cunning as mathematics. If you like to work a lot and make mistakes - you can continue to count your finger.

Well, to finally convince you that the degrees came up with loyards and cunnies to solve their life problems, not to create you problems, here are another couple of examples from life.

Example from life number 4

You have a million rubles. At the beginning of each year you earn every million another million. That is, every million will double at the beginning of each year. How much money will you have in the years? If you are sitting now and "you think your finger", then you are a very hardworking person and .. stupid. But most likely you will answer in a couple of seconds, because you are smart! So, in the first year - two multiplied two ... in the second year - what happened, another two, on the third year ... Stop! You noticed that the number multiplies itself. So, two in the fifth degree - a million! And now imagine that you have a competition and these million will receive the one who will find faster ... It is worth remembering the degree of numbers, what do you think?

Example from life number 5

You have a million. At the beginning of each year you earn each million two more. Great truth? Every million triples. How much money will you have after a year? Let's count. The first year is to multiply on, then the result is still on ... already boring, because you have already understood everything: three is multiplied by itself. Therefore, the fourth degree is equal to a million. It is only necessary to remember that three in the fourth degree is or.

Now you know that with the help of the erection of the number, you will greatly facilitate your life. Let's look next to what you can do with the degrees and what you need to know about them.

Terms and concepts ... so as not to get confused

So, for starters, let's define the concepts. What do you think, what is the indicator of the degree? It is very simple - this is the number that is "at the top" of the degree of number. Not scientifically, but it is clear and easy to remember ...

Well, at the same time that such a foundation degree? Even easier - this is the number that is below, at the base.

Here is a drawing for loyalty.

Well, B. generalTo summarize and better remember ... The degree with the foundation "" and the indicator "" is read as "to degree" and is written as follows:

The degree of number with a natural indicator

You already probably guessed: because the indicator is natural number. Yes, but what is natural number? Elementary! Natural These are the numbers that are used in the account when listing items: one, two, three ... We, when we consider items, do not say: "Minus five", "minus six", "minus seven". We also do not say: "one third", or "zero of whole, five tenths." These are not natural numbers. And what these numbers do you think?

Numbers like "minus five", "minus six", "minus seven" belong to whole numbers. In general, to whole numbers include all natural numbers, the numbers are opposite to natural (that is, taken with a minus sign), and the number. Zero understand easily - this is when nothing. And what do they mean negative ("minus") numbers? But they were invented primarily to designate debts: if you have a balance on the phone number, it means that you should operator rubles.

All sorts of fractions are rational numbers. How did they arise, what do you think? Very simple. Several thousand years ago, our ancestors found that they lack natural numbers to measure long, weight, square, etc. And they invented rational numbers... I wonder if it's true?

There are also irrational numbers. What is this number? If short, then an infinite decimal fraction. For example, if the circumference length is divided into its diameter, then the irrational number will be.

Summary:

We define the concept of degree, the indicator of which is a natural number (i.e., a whole and positive).

1. Any number to the first degree equally to itself:
2. Evaluate the number in the square - it means to multiply it by itself:
3. Evaluate the number in the cube - it means to multiply it by itself three times:

Definition. Evaluate the number in a natural degree - it means to multiply the number of all time for yourself:
.

Properties of degrees

Where did these properties come from? I will show you now.

Let's see: what is and ?

A-priory:

How many multipliers are here?

Very simple: we completed multipliers to multipliers, it turned out the factors.

But by definition, this is the degree of a number with an indicator, that is, that, that it was necessary to prove.

Example: Simplify the expression.

Decision:

Example: Simplify the expression.

Decision: It is important to notice that in our rule before Must be the same foundation!
Therefore, we combine degrees with the basis, but remains a separate multiplier:

only for the work of degrees!

In no case can not write that.

2. That is The degree of number

Just as with the previous property, we turn to the definition of the degree:

It turns out that the expression is multiplied by itself once, that is, according to the definition, this is, there is a number of number:

In fact, this can be called "the indicator for brackets". But never can do it in the amount:

Recall the formula of abbreviated multiplication: how many times did we want to write?

But it is incorrect, because.

Negative

Up to this point, we only discussed what the indicator should be.

But what should be the basis?

In the degrees of S. natural indicator The base can be any number. And the truth, we can multiply each other any numbers, whether they are positive, negative, or even.

Let's think about what signs ("or" ") will have the degrees of positive and negative numbers?

For example, a positive or negative number? BUT? ? With the first, everything is clear: how many positive numbers we are not multiplied by each other, the result will be positive.

But with negative a little more interesting. After all, we remember a simple rule of grade 6: "Minus for minus gives a plus." That is, or. But if we multiply on, it will work out.

Determine independently, what sign the following expressions will have:

1)	2)	3)
4)	5)	6)

Cope?

Here are the answers: in the first four examples, I hope everything is understandable? Just look at the base and indicator, and apply the appropriate rule.

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In example 5), everything is also not as scary, as it seems: it doesn't matter what is equal to the base - the degree is even, which means that the result will always be positive.

Well, with the exception of the case when the base is zero. The reason is not equal? Obviously no, because (because).

Example 6) is no longer so simple!

6 Examples for Training

Solutions of 6 examples

If you do not pay attention to the eighth degree, what do we see here? Remember the Grade 7 program. So, remembered? This is a formula for abbreviated multiplication, namely - the difference of squares! We get:

Carefully look at the denominator. He is very similar to one of the multipliers of the numerator, but what's wrong? Not the procedure of the terms. If they would change them in places, it would be possible to apply the rule.

But how to do that? It turns out very easy: the even degree of denominator helps us.

Magically, the components changed in places. This "phenomenon" is applicable for any expression to an even degree: we can freely change signs in brackets.

But it is important to remember: all signs are changing at the same time.!

Let's go back for example:

And again the formula:

Integer We call natural numbers opposite to them (that is, taken with the sign "") and the number.

whole positive number, And it does not differ from natural, then everything looks exactly as in the previous section.

And now let's consider new cases. Let's start with an indicator equal to.

Any number to zero equal to one:

As always, we will ask me: why is it so?

Consider any degree with the basis. Take, for example, and domineering on:

So, we multiplied the number on, and got the same as it was. And for what number must be multiplied so that nothing has changed? That's right on. So.

We can do the same with an arbitrary number:

Repeat the rule:

Any number to zero equal to one.

But from many rules there are exceptions. And here it is also there is a number (as a base).

On the one hand, it should be equal to any extent - how much zero itself is neither multiplied, still get zero, it is clear. But on the other hand, like any number to zero degree, should be equal. So what's the truth? Mathematics decided not to bind and refused to erect zero to zero. That is, now we can not only be divided into zero, but also to build it to zero.

Let's go further. In addition to natural numbers and numbers include negative numbers. To understand what a negative degree, we will do as last time: Domingly some normal number on the same to a negative degree:

From here it is already easy to express the desired:

Now we spread the resulting rule to an arbitrary degree:

So, we formulate the rule:

The number is a negative degree back to the same number to a positive degree. But at the same time the base can not be zero: (Because it is impossible to divide).

Let's summarize:

I. The expression is not defined in the case. If, then.

II. Any number to zero is equal to one :.

III. A number that is not equal to zero, to a negative degree back to the same number to a positive degree :.

Tasks for self solutions:

Well, as usual, examples for self solutions:

Task analysis for self solutions:

I know, I know, the numbers are terrible, but the exam should be ready for everything! Share these examples or scatter their decision, if I could not decide and you will learn to easily cope with them on the exam!

Continue expanding the circle of numbers, "suitable" as an indicator of the degree.

Now consider rational numbers. What numbers are called rational?

Answer: All that can be represented in the form of fractions, where and - integers, and.

To understand what is "Freight degree", Consider the fraction:

Erected both parts of the equation to the degree:

Now remember the rule about "Degree to degree":

What number should be taken to the degree to get?

This formulation is the definition of root degree.

Let me remind you: the root of the number () is called the number that is equal in the extermination.

That is, the root degree is an operation, reverse the exercise into the degree :.

Turns out that. Obviously, this particular case can be expanded :.

Now add a numerator: what is? The answer is easy to get with the help of the "degree to degree" rule:

But can the reason be any number? After all, the root can not be extracted from all numbers.

No one!

Remember the rule: any number erected into an even degree is the number positive. That is, to extract the roots of an even degree from negative numbers it is impossible!

This means that it is impossible to build such numbers into a fractional degree with an even denominator, that is, the expression does not make sense.

What about expression?

But there is a problem.

The number can be represented in the form of DRGIH, reduced fractions, for example, or.

And it turns out that there is, but does not exist, but it's just two different records of the same number.

Or another example: once, then you can write. But it is worthwhile to write to us in a different way, and again we get a nuisance: (that is, they received a completely different result!).

To avoid similar paradoxes, we consider only a positive foundation of degree with fractional indicator.

So, if:

• - natural number;
• - integer;

Examples:

The degrees with the rational indicator are very useful for converting expressions with roots, for example:

5 examples for training

Analysis of 5 examples for training

Well, now - the most difficult. Now we will understand irrational.

All the rules and properties of degrees here are exactly the same as for a degree with a rational indicator, with the exception

After all, by definition, irrational numbers are numbers that cannot be represented in the form of a fraction, where and - integers (that is, irrational numbers are all valid numbers except rational).

When studying degrees with natural, whole and rational indicator, we each time constituted a certain "image", "analogy", or a description in more familiar terms.

For example, a natural figure is a number, several times multiplied by itself;

...zero - this is how the number multiplied by itself once, that is, it has not yet begun to multiply, it means that the number itself has not even appeared - therefore the result is only a certain "billet number", namely the number;

...degree with a whole negative indicator "It seemed to have occurred a certain" reverse process ", that is, the number was not multiplied by itself, but Deli.

By the way, in science is often used with a complex indicator, that is, the indicator is not even a valid number.

But at school we do not think about such difficulties, you will have the opportunity to comprehend these new concepts at the Institute.

Where we are sure you will do! (If you learn to solve such examples :))

For example:

Solim yourself:

Debris:

1. Let's start with the usual rules for the exercise rules for us:

Now look at the indicator. Doesn't he remind you of anything? Remember the formula of abbreviated multiplication. Square differences:

In this case,

Turns out that:

Answer: .

2. We bring the fraction in the indicators of degrees to the same form: either both decimal or both ordinary. We obtain, for example:

Answer: 16.

3. Nothing special, we use the usual properties of degrees:

ADVANCED LEVEL

Determination of degree

The degree is called the expression of the form: where:

• — degree basis;
• - Indicator.

The degree with the natural indicator (n \u003d 1, 2, 3, ...)

Build a natural degree n - it means multiplying the number for yourself once:

The degree with the integer (0, ± 1, ± 2, ...)

If an indicator of the degree is software positive number:

Construction in zero degree:

The expression is indefinite, because, on the one hand, to any extent, it is, and on the other - any number of in degree is.

If an indicator of the degree is a whole negative number:

(Because it is impossible to divide).

Once again about zeros: the expression is not defined in the case. If, then.

Examples:

Rational

• - natural number;
• - integer;

Examples:

Properties of degrees

To make it easier to solve problems, let's try to understand: where did these properties come from? We prove them.

Let's see: What is what?

A-priory:

So, in the right part of this expression, such a work is obtained:

But by definition, this is the degree of a number with an indicator, that is:

Q.E.D.

Example : Simplify the expression.

Decision : .

Example : Simplify the expression.

Decision : It is important to notice that in our rule beforethere must be the same bases. Therefore, we combine degrees with the basis, but remains a separate multiplier:

Another important note: this is a rule - only for the work of degrees!

In no case to the nerve to write that.

Just as with the previous property, we turn to the definition of the degree:

We regroup this work like this:

It turns out that the expression is multiplied by itself once, that is, according to the definition, this is - by the degree of number:

In fact, this can be called "the indicator for brackets". But never can do this in the amount:!

Recall the formula of abbreviated multiplication: how many times did we want to write? But it is incorrect, because.

Degree with a negative basis.

Up to this point, we only discussed what should be indicator degree. But what should be the basis? In the degrees of S. natural indicator The base can be any number .

And the truth, we can multiply each other any numbers, whether they are positive, negative, or even. Let's think about what signs ("or" ") will have the degrees of positive and negative numbers?

For example, a positive or negative number? BUT? ?

With the first, everything is clear: how many positive numbers we are not multiplied by each other, the result will be positive.

But with negative a little more interesting. After all, we remember a simple rule of grade 6: "Minus for minus gives a plus." That is, or. But if we will multiply on (), it turns out.

And so to infinity: each time the next multiplication will change the sign. Simple rules can be formulated:

1. even degree - number positive.
2. Negative number erected into odd degree - number negative.
3. A positive number to either degree is the number positive.
4. Zero to any degree is zero.

Determine independently, what sign the following expressions will have:

1.	2.	3.
4.	5.	6.

Cope? Here are the answers:

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In the first four examples, I hope everything is clear? Just look at the base and indicator, and apply the appropriate rule.

In example 5), everything is also not as scary, as it seems: it doesn't matter what is equal to the base - the degree is even, which means that the result will always be positive. Well, with the exception of the case when the base is zero. The reason is not equal? Obviously no, because (because).

Example 6) is no longer so simple. Here you need to know that less: or? If you remember that it becomes clear that, and therefore, the base is less than zero. That is, we apply the rule 2: the result will be negative.

And again we use the degree of degree:

All as usual - write down the definition of degrees and, divide them to each other, divide on the pairs and get:

Before you disassemble the last rule, we solve several examples.

Calculated expressions:

Solutions :

If you do not pay attention to the eighth degree, what do we see here? Remember the Grade 7 program. So, remembered? This is a formula for abbreviated multiplication, namely - the difference of squares!

We get:

Carefully look at the denominator. He is very similar to one of the multipliers of the numerator, but what's wrong? Not the procedure of the terms. If they were swapped in places, it would be possible to apply the rule 3. But how to do it? It turns out very easy: the even degree of denominator helps us.

If you draw it on, nothing will change, right? But now it turns out the following:

Magically, the components changed in places. This "phenomenon" is applicable for any expression to an even degree: we can freely change signs in brackets. But it is important to remember: all signs are changing at the same time!You can not replace on, changing only one disagreeable minus!

Let's go back for example:

And again the formula:

So now the last rule:

How will we prove? Of course, as usual: I will reveal the concept of degree and simplifies:

Well, now I will reveal brackets. How much will the letters get? Once on multipliers - what does it remind? It is nothing but the definition of the operation multiplication: In total there were factors. That is, it is, by definition, the degree of number with the indicator:

Example:

Irrational

In addition to information about degrees for the average level, we will analyze the degree with the irrational indicator. All the rules and properties of degrees here are exactly the same as for a degree with a rational indicator, with the exception - after all, by definition, irrational numbers are numbers that cannot be submitted in the form of a fraction, where - the integers (i.e., irrational numbers are All valid numbers besides rational).

When studying degrees with natural, whole and rational indicator, we each time constituted a certain "image", "analogy", or a description in more familiar terms. For example, a natural figure is a number, several times multiplied by itself; The number in zero degree is somehow the number multiplied by itself once, that is, it has not yet begun to multiply, it means that the number itself has not even appeared - therefore, only a certain "billet", namely, is the result; The degree with a whole negative indicator is as if a certain "reverse process" occurred, that is, the number was not multiplied by itself, but divided.

Imagine the degree with an irrational indicator is extremely difficult (just as it is difficult to submit a 4-dimensional space). It is rather clean mathematical objectwhich mathematics created to expand the concept of degree to the entire number of numbers.

By the way, in science is often used with a complex indicator, that is, the indicator is not even a valid number. But at school we do not think about such difficulties, you will have the opportunity to comprehend these new concepts at the Institute.

So what do we do if we see an irrational rate? We are trying to get rid of it with all the might! :)

For example:

Solim yourself:

1)

2)

3)

Answers:

1. We remember the formula the difference of squares. Answer:.
2. We give the fraction to the same form: either both decimal, or both ordinary. We get, for example:.
3. Nothing special, we use the usual properties of degrees:

Summary of section and basic formulas

Degree called the expression of the form: where:

Integer

the degree, the indicator of which is a natural number (i.e., a whole and positive).

Rational

the degree, the indicator of which is negative and fractional numbers.

Irrational

the degree, the indicator of which is an infinite decimal fraction or root.

Properties of degrees

Features of degrees.

• Negative number erected into even degree - number positive.
• Negative number erected into odd degree - number negative.
• A positive number to either degree is the number positive.
• Zero to any degree is equal.
• Any number to zero equal.

Now you need a word ...

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Tell me about your experience in using the properties of degrees.

Perhaps you have questions. Or suggestions.

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And good luck on the exams!