At the beginning of the lesson we will repeat the basic properties square rootsand then consider a few complex examples To simplify expressions containing square roots.

Subject:Function. Properties of square root

Lesson:Transformation and simplification of more complex expressions with roots

1. Repeat the properties of square roots

Briefly repeat the theory and remind the basic properties of square roots.

Properties of square roots:

1., therefore;

3. ;

4. .

2. Examples for simplifying expressions with roots

Let us turn to the examples of using these properties.

Example 1. Simplify expression .

Decision. To simplify the number 120, it is necessary to decompose on simple factors:

Square amount will be revealed according to the corresponding formula:

Example 2. Simplify expression .

Decision. We take into account that this expression makes sense not with all possible values \u200b\u200bof the variable, since there is square roots and fractions, which leads to the "narrowing" of the area of \u200b\u200bpermissible values. OTZ: ().

We give the expression in brackets to the general denominator and with a spinner of the last fraction as a difference of squares:

Answer. at.

Example 3. Simplify expression .

Decision. It can be seen that the second numerator bracket has an uncomfortable look and needs to be simplified, try to decompose it for multipliers using the grouping method.

For the possibility of making a common factor, we simplified the roots by decomposition of multipliers. We substitute the resulting expression in the original fraction:

After cutting the fraction, apply the formula of the difference in squares.

3. Example on getting rid of irrationality

Example 4. Frequently from irrationality (roots) in the denominator: a); b).

Decision. a) in order to get rid of irrationality in the denominator, applied standard method Domination and numerator and denominator of the fraction on the factor conjugate to the denominator (the same expression, but with a reverse sign). This is done to supplement the denominator of the fraction to the difference in squares, which allows the root from the root in the denominator. Perform this technique in our case:

b) Perform similar actions:

4. Example for evidence and on the release of a full square in a complex radical

Example 5. Prove equality .

Evidence. We use the definition of the square root, from which it follows that the square of the right expression should be equal to the guided expression:

. We will reveal brackets by the Square Formula:

Required true equality.

Proved.

Example 6. Simplify the expression.

Decision. The specified expression is customary called a complex radical (root under the root). IN this example It is necessary to guess to allocate a full square from the feeding expression. To do this, we note that of the two components is a challenge for the role of a doubled work in the formula of the square of the difference (difference, since there is a minus). We bring it in the form of such a work:, then the role of one of the complications of the full square is claimed, and on the role of the second - 1.

We will substitute this expression on the root.

An alpoint expression (or expression with variables) is a mathematical expression that consists of numbers, letters and signs of mathematical operations. For example, the following expression is alphabetic:

a + b + 4

With the help of letter expressions, you can record laws, formulas, equations and functions. The ability to manipulate letterproof expressions is the key to good knowledge of algebra and higher mathematics.

Any serious task in mathematics is reduced to solving equations. And to be able to solve equations, you need to be able to work with the letter expressions.

To work with letterproof expressions, you need to study well the basic arithmetic: addition, subtraction, multiplication, division, basic laws of mathematics, fractions, action with fractions, proportions. And not just explore, but to understand thoroughly.

Design of lesson

Variables

Letters that are contained in alphabet expressions are called variables. For example, in expression a + b + 4 Changes are letters a. and b.. If instead of these variables, substitute any numbers, then the letter expression a + b + 4 Contact the numeric expression whose value can be found.

Numbers that are substituted instead of variables call values \u200b\u200bof variables. For example, change the values \u200b\u200bof variables a. and b.. To change the values, an equal sign is used

a \u003d 2, b \u003d 3

We changed the values \u200b\u200bof variables a. and b.. Variable a. Assigned importance 2 , variable b. Assigned importance 3 . As a result, an alphabetic expression a + b + 4 appeals to the usual numerical expression 2+3+4 whose value can be found:

2 + 3 + 4 = 9

When multiplication of variables occurs, they are recorded together. For example, writing aB means the same as recording a × B.. If we substitute instead of variables a.and B. numbers 2 and 3 then we will get 6

2 × 3 \u003d 6

It can also be supposed to comply with the multiplication of the number on the expression in brackets. For example, instead a × (B + C) can be recorded a (B + C). Applying the distribution law of multiplication, we get a (B + C) \u003d AB + AC.

Factors

In letter expressions, you can often find a record in which the number and variable are recorded together, for example 3A . In fact, this is a short recording of multiplication of the number 3 to the variable a. And this entry looks like 3 × A. .

In other words, expression 3A It is a product of the number 3 and variable a.. Number 3 In this work they call coefficient. This coefficient shows how many times the variable will be increased. a.. This expression can be read as " a. three times "or" three times but", Or" increase the value of the variable a. three times ", but most often read as" three a.«

For example, if the variable a. equal 5 then the value of the expression 3Ait will be 15.

3 × 5 \u003d 15

Speaking simple language, The coefficient is a number that is facing the letter (before the variable).

Letters can be somewhat, for example 5ABc.. Here the coefficient is the number 5 . This coefficient shows that the product of variables aBC Increases five times. This expression can be read as " aBC five times "either" increase the expression value aBC five times "or" five aBC«.

If instead instead of variables aBC substitute numbers 2, 3 and 4, then the value of the expression 5ABc. will be equal 120

5 × 2 × 3 × 4 \u003d 120

You can mentally imagine how first the numbers 2, 3 and 4 are meditated, and the resulting value increased five times:

The coefficient mark applies only to the coefficient, and does not relate to variables.

Consider expression -6B.. Minus factor 6 , applies only to the coefficient 6 and does not apply to the variable b.. Understanding this fact will not make mistakes in the future with signs.

Find the value of the expression -6B. for b \u003d 3..

-6B. -6 × B.. For clarity, write the expression -6B. in deployment and substitute the value of the variable b.

-6b \u003d -6 × b \u003d -6 × 3 \u003d -18

Example 2. Find an expression value -6B. for b \u003d -5

We write expression -6B. in deployed video

-6b \u003d -6 × b \u003d -6 × (-5) \u003d 30

Example 3. Find an expression value -5A + B. for a \u003d 3.and B \u003d 2.

-5A + B. This is a short form of recording from -5 × a + b , so for clarity we write the expression -5 × a + b in deployment and substitute the values \u200b\u200bof variables a. and b.

-5A + B \u003d -5 × A + B \u003d -5 × 3 + 2 \u003d -15 + 2 \u003d -13

Sometimes letters are written without a coefficient, for example a. or aB . In this case, the coefficient is a unit:

but the unit according to tradition is not recorded, so they just write a. or aB

If there is a minus before the letter, then the coefficient is the number −1 . For example, expression -A. actually looks like -1A.. This is a product minus units and variable a.It was as follows:

-1 × a \u003d -1a

Here lies a little catch. In expression -A. minus facing variable a. in fact refers to the "invisible unit", and not to the variable a. . Therefore, when solving tasks should be attentive.

For example, if the expression is given -A. and we are asked to find its meaning when a \u003d 2. then in school we substituted a two instead of a variable a. and received the answer −2 , not particularly documented on how it turned out. In fact, there was a multiplication of minus units on a positive number 2

-A \u003d -1 × a

-1 × a \u003d -1 × 2 \u003d -2

If the expression is given -A. and it is required to find its meaning when a \u003d -2. then we substitute −2 Instead of a variable a.

-A \u003d -1 × a

-1 × a \u003d -1 × (-2) \u003d 2

To avoid errors, the first time invisible units can be written explicitly.

Example 4. Find an expression value aBC for a \u003d 2. , b \u003d 3. and c \u003d 4.

Expression aBC 1 × a × b × c. For clarity, write the expression aBC a, B. and c.

1 × A × B × C \u003d 1 × 2 × 3 × 4 \u003d 24

Example 5. Find an expression value aBC for a \u003d -2, b \u003d -3and C \u003d -4.

We write expression aBC in deployment and substitute the values \u200b\u200bof variables a, B.and C.

1 × a × b × c \u003d 1 × (-2) × (-3) × (-4) \u003d -24

Example 6. Find an expression value − aBC for a \u003d 3, b \u003d 5 and c \u003d 7

Expression − aBC This is a short form of recording from -1 × a × b × c. For clarity, write the expression − aBC in deployment and substitute the values \u200b\u200bof variables a, B. and c.

-Abc \u003d -1 × a × b × C \u003d -1 × 3 × 5 × 7 \u003d -105

Example 7. Find an expression value − aBC for a \u003d -2, b \u003d4 and c \u003d -3

We write expression − aBC In an expanded form:

-Abc \u003d -1 × a × b × c

We substitute the value of variables a. , b. and c.

-Abc \u003d -1 × a × b × c \u003d -1 × (-2) × (-4) × (-3) \u003d 24

How to determine the coefficient

Sometimes it is required to solve the task in which the expression coefficient is required. In principle, this task is very simple. It is enough to be able to correctly multiply the numbers.

To determine the coefficient in the expression, it is necessary to multiply the numbers included in this expression, and multiply multiply the letters. The resulting numerical factory and will be a coefficient.

Example 1. 7m × 5a × (-3) × n

The expression consists of several factors. It can be clearly seen if you write an expression in the deployment. That is, works 7m. and 5a Record in the form 7 × M. and 5 × A.

7 × m × 5 × a × (-3) × n

We apply a combination of multiplication law, which allows multipliers to multiply in any order. Namely, separately change the numbers and separately with the letters (variables):

-3 × 7 × 5 × m × a × n \u003d -105man

The coefficient is equal −105 . After completion, the letter part is desirable to arrange in alphabetical order:

-105amn.

Example 2. Determine the coefficient in expression: -A × (-3) × 2

-A × (-3) × 2 \u003d -3 × 2 × (-a) \u003d -6 × (-a) \u003d 6A

The coefficient is 6.

Example 3. Determine the coefficient in expression:

Move separately numbers and letters:

The coefficient is -1. Note that the unit is not recorded because the coefficient 1 is not recorded not to record.

These seemingly the simplest tasks can play with us a very evil joke. It often finds out that the coefficient's sign is incorrect: either missed minus or vice versa it is in vain. To avoid these annoying errors, should be studied at a good level.

Consciousness in alphabetic expressions

When adding several numbers, the sum of these numbers is obtained. The numbers that are called called the terms. The components can be several, for example:

1 + 2 + 3 + 4 + 5

When the expression consists of the components, it is much easier to calculate it, because it is easier to add than to deduct. But in the expression there may be not only addition, but also subtraction, for example:

1 + 2 − 3 + 4 − 5

In this expression, the number 3 and 5 are subtracted, and not the terms. But it does not interfere with us, replace subtraction by adding. Then we again get an expression consisting of the terms:

1 + 2 + (−3) + 4 + (−5)

Do not escape that numbers -3 and -5 now with a minus sign. The main thing is that all numbers in this expression are connected by the addition mark, that is, the expression is the amount.

Both expressions 1 + 2 − 3 + 4 − 5 and 1 + 2 + (−3) + 4 + (−5) equal to one and that value - minus one

1 + 2 − 3 + 4 − 5 = −1

1 + 2 + (−3) + 4 + (−5) = −1

Thus, the value of the expression does not suffer from the fact that we will replace subtracting by adding.

Replacing subtraction by adding can also be in alphabone expressions. For example, consider the following expression:

7A + 6B - 3C + 2D - 4S

7A + 6B + (-3c) + 2D + (-4S)

With any values \u200b\u200bof variables a, B, C, D and s. Expressions 7A + 6B - 3C + 2D - 4S and 7A + 6B + (-3c) + 2D + (-4S) will be equal to the same value.

You must be prepared for the fact that the teacher at school or the teacher at the institute can call the alignments even those numbers (or variables) that they are not.

For example, if a difference will be recorded on the board a - B. then the teacher will not say that a. - this is diminished, and b. - subtracted. Both variables, he will call one general word - composition . And all because the expression of the form a - B. Mathematics sees how the amount a + (-B) . In this case, the expression becomes the amount, and variables a. and (-B) become the terms.

Similar terms

Similar terms - These are the terms that have the same alphabetic part. For example, consider expression 7A + 6B + 2A . Composition 7A. and 2a. have the same alphabet part - variable a.. So the components 7A. and 2a.are similar.

Usually, the similar components are folded to simplify the expression or solve some equation. This operation is called by bringing similar terms.

To bring similar terms, you need to fold the coefficients of these terms, and the resulting result is multiplied by the overall letter.

For example, we give similar terms in the expression 3A + 4A + 5A . In this case, these are all the terms. Moving their coefficients and the result to multiply on the general lettering part - to the variable a.

3A + 4A + 5A \u003d (3 + 4 + 5) × a \u003d 12a

Similar terms usually lead in mind and the result is recorded immediately:

3A + 4A + 5A \u003d 12A

Also, one can argue as follows:

There were 3 variables A, they added another 4 variables A and 5 more variables a. As a result, 12 variables were obtained

Consider several examples of bringing similar terms. Considering that this topic It is very important, at first we will write in detail every little thing. Despite the fact that everything is very simple here, most people allow many mistakes. Basically intensifying, and not for ignorance.

Example 1. 3A + 2A + 6A + 8a.

Moving the coefficients in this expression and the result obtained to multiply on the general lettering part:

3A + 2A + 6A + 8A \u003d (3 + 2 + 6 + 8) × a \u003d 19A

Design (3 + 2 + 6 + 8) × a You can not record, so you will immediately write the answer

3A + 2A + 6A + 8A \u003d 19A

Example 2. Create similar components in expression 2A + A.

Second term a. recorded without a coefficient, but in fact there is a coefficient 1 which we do not see because of the fact that it is not written. Therefore, the expression looks like this:

2A + 1A.

Now we give similar terms. That is, lay the coefficients and the result to multiply on the general lettering:

2a + 1a \u003d (2 + 1) × a \u003d 3a

Write the decision shorter:

2a + a \u003d 3a

2A + A., it can be reasoned and different:

Example 3. Create similar components in expression 2A - A.

Replace subtraction by adding:

2a + (-a)

Second term (-A) written without a coefficient, but it looks like (-1A).Coefficient −1 Again, invisible due to the fact that it is not written. Therefore, the expression looks like this:

2a + (-1a)

Now we give similar terms. Mix the coefficients and the result to multiply on the overall letter:

2a + (-1a) \u003d (2 + (-1)) × a \u003d 1a \u003d a

Usually recorded in short:

2a - a \u003d a

Leading similar components in expression 2a-A. It can be reasoned in a different way:

There were 2 variables a, detected one variable A, as a result, one single variable remained

Example 4. Create similar components in expression 6A - 3A + 4A - 8A

6A - 3A + 4A - 8A \u003d 6A + (-3A) + 4A + (-8A)

Now we give similar terms. Mix the coefficients and the result to multiply on the overall letter

(6 + (-3) + 4 + (-8)) × a \u003d -1a \u003d -a

Write the decision shorter:

6A - 3A + 4A - 8A \u003d -A

There are expressions that contain several different groups of similar terms. For example, 3A + 3B + 7A + 2B . For such expressions, the same rules are valid as for the rest, namely the folding of the coefficients and the multiplication of the result obtained on the overall letter. But in order to prevent errors, it is convenient to emphasize different lines of components.

For example, in expression 3A + 3B + 7A + 2B those terms that contain a variable a., you can emphasize with one line, and those components that contain a variable b., you can emphasize two lines:

Now you can bring similar terms. That is, fold the coefficients and the resulting result is multiplied by the general letter. This is necessary for both groups of terms: for terms containing a variable a. And for the components containing the variable b..

3A + 3B + 7A + 2B \u003d (3 + 7) × A + (3 + 2) × B \u003d 10A + 5B

Again, we repeat, the expression is simple, and the like components can be given in mind:

3A + 3B + 7A + 2B \u003d 10A + 5B

Example 5. Create similar components in expression 5A - 6A -7B + B

Replace subtraction addition where it can be:

5a - 6A -7B + B \u003d 5A + (-6A) + (-7B) + B

We emphasize the similar terms of different lines. Condivables containing variables a. We emphasize one line, and the components of the variables b. , we emphasize the two lines:

Now you can bring similar terms. That is, fold the coefficients and the result obtained multiplied by the general letter:

5a + (-6a) + (-7b) + b \u003d (5 + (-6)) × a + ((-7) + 1) × b \u003d -a + (-6b)

If the expression contains ordinary numbers without alleged factors, they add up separately.

Example 6. Create similar components in expression 4A + 3A - 5 + 2B + 7

Replace subtraction by adding where it can be:

4A + 3A - 5 + 2B + 7 \u003d 4A + 3A + (-5) + 2B + 7

We give similar terms. Numbers −5 and 7 do not have alphabets, but they are similar terms - they need to just fold. And the foundation 2b. will remain unchanged because it is the only thing in this expression that has an alphabet b, And nothing to fold it with:

4a + 3a + (-5) + 2b + 7 \u003d (4 + 3) × A + 2B + (-5) + 7 \u003d 7A + 2B + 2

Write the decision shorter:

4A + 3A - 5 + 2B + 7 \u003d 7A + 2B + 2

The components can be organized so that those terms that have the same alphabetic part are located in one part of the expression.

Example 7. Create similar components in expression 5T + 2X + 3X + 5T + X

Since the expression is the sum of several terms, it allows us to calculate it in any order. Therefore, the components containing a variable t. can be written at the beginning of the expression, and the components containing the variable x. At the end of the expression:

5T + 5T + 2x + 3x + x

Now you can bring similar terms:

5t + 5t + 2x + 3x + x \u003d (5 + 5) × t + (2 + 3 + 1) × x \u003d 10t + 6x

Write the decision shorter:

5T + 2X + 3X + 5T + X \u003d 10T + 6X

The sum of the opposite numbers is zero. This rule works for alphabetic expressions. If the expression will meet the same terms, but with opposite signs, then they can be rid of them at the stage of bringing similar terms. In other words, simply shut out them from the expression, since their sum is zero.

Example 8. Create similar components in expression 3T - 4T - 3T + 2T

Replace subtraction by adding where it can be:

3T - 4T - 3T + 2T \u003d 3T + (-4t) + (-3t) + 2t

Composition 3T and (-3t) are opposite. The sum of the opposite terms is zero. If you remove this zero from the expression, the value of the expression does not change, so we will remove it. And we will remove it with the usual strikeout of the terms 3T and (-3t)

As a result, we will have an expression (-4t) + 2t. In this expression, such a component can be given and get the final answer:

(-4t) + 2t \u003d ((-4) + 2) × t \u003d -2t

Write the decision shorter:

Simplification of expressions

"Similarize the expression" And then the expression is given to simplify. Simplify expression So make it easier and shorter.

In fact, we have already been simplified expressions when the fractions have shrink. After cutting, the fraction became shorter and easier for perception.

Consider next example. Simplify the expression.

This task can literally understand this way: "Apply any permissible actions to this expression, but make it easier." .

In this case, the fraction can be reduced, namely split the numerator and denominator of the fraction 2:

What else can you do? You can calculate the resulting fraction. Then we get a decimal fraction 0.5

As a result, the fraction simply simplified to 0.5.

The first question that needs to be asked to solve such tasks should be "What can I do?" . Because there are actions that can be done, and there are actions that can not be done.

Another important momentWhat needs to be remembered is that the expression value should not change after simplifying the expression. Let's return to expression. This expression is a division that can be performed. By doing this division, we obtain the value of this expression, which is 0.5

But we simplified the expression and got a new simplified expression. The value of the new simplified expression is still 0.5

But the expression we also tried to simplify, calculate it. As a result, they received the final answer 0.5.

Thus, no matter how we simplify the expression, the value of the expressions obtained is still 0.5. So simplification was performed correctly at each stage. It is for this that it is necessary to strive when simplifying expressions - the value of the expression should not suffer from our actions.

Often it is necessary to simplify letter expressions. For them, the same facilities are fair as for numerical expressions. You can perform any permissible actions, just not to change the value of the expression.

Consider several examples.

Example 1. Simplify expression 5,21s × t × 2.5

To simplify this expression, you can multiply the numbers separately and multiply the letters. This task is very similar to the one we have considered when they learned to determine the coefficient:

5,21s × t × 2,5 \u003d 5.21 × 2.5 × s × t \u003d 13,025 × ST \u003d 13,025ST

So the expression 5,21s × t × 2.5 Simplified before 13,025ST.

Example 2. Simplify expression -0.4 × (-6,3b) × 2

Second work (-6,3b) can be translated to understandable to us, namely, write in the form ( -6.3) × b,then send the numbers separately and multiply the letters separately:

− 0,4 × (-6,3b) × 2 = − 0,4 × (-6.3) × b × 2 \u003d 5,04b

So the expression -0.4 × (-6,3b) × 2 Simplified before 5,04b

Example 3. Simplify expression

Cut this expression in more detail to see well where numbers, and where letters:

Now separately alternate numbers and separately alternate the letters:

So the expression Simplified before -ABC.This solution can be written shorter:

When simplifying expressions, the fraction can be reduced during the solution, and not at the very end, as we did it with ordinary fractions. For example, if during the solution we observe the expression of the form, then it is not necessary to calculate the numerator and the denominator and do something like this:

The fraction can be reduced by choosing in a multiplier in a numerator and in the denominator and cut these factors to their largest common divisor. In other words, to use in which we do not paint in detail what the numerator and denominator were divided.

For example, in the numerator multiplier 12 and in the denominator, the multiplier 4 can be reduced by 4. The fourth is stored in the mind, and dividing 12 and 4 to this fourth, the answers are recorded next to these numbers, after after following them

Now you can multiply the resulting small multipliers. In this case, they are a bit and can multiply in the mind:

Over time, it can be found that solving one or another task, expressions begin to "fat", so it is desirable to learn to rapid calculations. What can be calculated in the mind must be calculated in the mind. What you can quickly cut, you need to quickly cut.

Example 4. Simplify expression

So the expression Simplified before

Example 5. Simplify expression

Move separately the numbers and separate letters:

So the expression Simplified before mn.

Example 6. Simplify expression

We write this expression in more detail to see well where numbers, and where letters:

Now separately alternate the number and separate letters. For the convenience of computing decimal fraction -6.4 and mixed number You can translate into ordinary fractions:

So the expression Simplified before

The solution for this example can be recorded significantly shorter. It will look like this:

Example 7. Simplify expression

Move separately the numbers and separate letters. For convenience of calculating a mixed number and decimal fractions 0.1 and 0.6 can be translated into ordinary fractions:

So the expression Simplified before abcd.. If you skip the details, this decision can be recorded significantly in short:

Pay attention to how the fraction has decreased. New multipliers that are obtained as a result of the reduction of previous multipliers are also allowed to reduce.

Now let's talk about what you can not do. When simplifying expressions, it is categorically impossible to multiply the numbers and letters, if the expression is the sum, and not by the work.

For example, if you need to simplify expression 5A + 4B.You can not write as follows:

It is equivalent to the fact that if we were asked to fold two numbers, and we would multiply them instead of folding.

When substituting any values \u200b\u200bof variables a. and b. expression 5A + 4B. refers to an ordinary numerical expression. Suppose that variables a. and b. have the following values:

a \u003d 2, b \u003d 3

Then the expression value will be equal to 22

5A + 4B \u003d 5 × 2 + 4 × 3 \u003d 10 + 12 \u003d 22

First, multiplication is performed, and then the results are folded. And if we tried to simplify this expression, moving the numbers and letters, it would have happened:

5A + 4B \u003d 5 × 4 × a × b \u003d 20ab

20ab \u003d 20 × 2 × 3 \u003d 120

It turns out a completely different value of the expression. In the first case it turned out 22 in the second case 120 . This means that simplification of expression 5A + 4B. It was incorrect.

After simplifying the expression, its value should not be changed at the same values \u200b\u200bof variables. If during substitution to the initial expression of any values \u200b\u200bof variables, one value is obtained, then after simplifying the expression, the same value should be obtained as before simplification.

With an expression 5A + 4B. In fact, you can not do anything. It is not simplified.

If the expression contains similar components, they can be folded if our goal is to simplify expression.

Example 8. Simplify expression 0,3A-0,4A + a

0,3A - 0,4A + a \u003d 0,3a + (-0.4a) + a \u003d (0.3 + (-0.4) + 1) × a \u003d 0,9A

or shorter: 0,3A - 0,4A + A = 0.9A.

So the expression 0,3A-0,4A + a Simplified before 0.9A.

Example 9. Simplify expression -7,5A - 2.5B + \u200b\u200b4A

To simplify this expression, you can bring similar terms:

-7,5A - 2.5B + \u200b\u200b4a \u003d -7,5A + (-2,5b) + 4a \u003d ((-7.5) + 4) × A + (-2,5B) \u003d -3,5A + (-2,5B)

or shorter -7,5A - 2.5B + \u200b\u200b4A \u003d -3,5A + (-2,5B)

Speed (-2,5B) It remains unchanged, because it has nothing to be folded.

Example 10. Simplify expression

To simplify this expression, you can bring similar terms:

The coefficient was for the convenience of calculating.

So the expression Simplified before

Example 11. Simplify expression

To simplify this expression, you can bring similar terms:

So the expression Simplified before.

In this example, it would be more expedient to fold the first and last coefficient in the first place. In this case, we would get a short decision. It looked as follows:

Example 12. Simplify expression

To simplify this expression, you can bring similar terms:

So the expression Simplified before .

The term remained unchanged, because it has nothing to be folded.

This solution can be recorded significantly shorter. It will look like this:

IN short decision The stages of replacement of subtraction by addition and the detailed entry, as the fraction was brought to a common denominator.

Another distinction is that detailed decision The answer looks like , and in short as. In fact, this is the same expression. The difference is that in the first case, subtraction is replaced by adding, because at the beginning when we recorded the decision in detailedWe are everywhere where you can replace subtraction by adding, and this replacement has been preserved for answering.

Identities. Identically equal expressions

After we simplified any expression, it becomes easier and shorter. To check whether the expression is simplified, it is enough to substitute any values \u200b\u200bof variables first into the previous expression that was required to simplify, and then to the new one that was simplified. If the value in both expressions is the same, the expression is simplified correctly.

Consider the simplest example. Let it take to simplify the expression 2a × 7b. . To simplify this expression, you can multiply numbers and letters separately:

2a × 7b \u003d 2 × 7 × a × b \u003d 14ab

Check whether we simplified the expression. To do this we will substitute any values \u200b\u200bof variables a. and b. First, in the first expression that was required to simplify, and then second, which was simplified.

Let the values \u200b\u200bof the variables a. , b. will be as follows:

a \u003d 4, b \u003d 5

Substitute them in the first expression 2a × 7b.

Now we will substitute the same values \u200b\u200bof variables in the expression that happened as a result of simplification 2a × 7b., namely, the expression 14ab

14ab \u003d 14 × 4 × 5 \u003d 280

We see that when a \u003d 4. and b \u003d 5. The value of the first expression 2a × 7b. and the value of the second expression 14ab equal

2a × 7b \u003d 2 × 4 × 7 × 5 \u003d 280

14ab \u003d 14 × 4 × 5 \u003d 280

The same thing will happen for any other values. For example, let it a \u003d 1. and b \u003d 2.

2a × 7b \u003d 2 × 1 × 7 × 2 \u003d 28

14ab \u003d 14 × 1 × 2 \u003d 28

Thus, with any values \u200b\u200bof variable expression 2a × 7b. and 14ab equal to the same meaning. Such expressions are called identically equal.

We conclude that between expressions 2a × 7b. and 14ab You can put a sign of equality, since they are equal to the same value.

2a × 7b \u003d 14ab

The equality is called any expression that is connected by the sign of equality (\u003d).

A equality of type 2a × 7b \u003d 14ab Call identity.

The identity is called equality that is true for any values \u200b\u200bof variables.

Other examples of identities:

a + b \u003d b + a

a (B + C) \u003d AB + AC

a (BC) \u003d (AB) C

Yes, the laws of mathematics, which we studied are identities.

Faithful numeric equality are also identities. For example:

2 + 2 = 4

3 + 3 = 5 + 1

10 = 7 + 2 + 1

Solving a complex task to facilitate the calculation, complex expression Replace on a simpler expression identically equal to the previous one. Such a replacement is called identical transformation of expression or simply transformation of the expression.

For example, we have simplified expression 2a × 7b. and got a simpler expression 14ab . This simplification can be called identical conversion.

Often you can meet the task in which it is said "Prove that equality is the identity" And then the equality that needs to be proved is given. Typically, this equality consists of two parts: the left and right part of equality. Our task is to perform identical conversions with one of the parts of equality and get another part. Either perform identical transformations with both parts of equality and make such an equal expression in both parts of equality.

For example, we prove that equality 0,5A × 5B \u003d 2,5ab Is identity.

We simplify the left part of this equality. To do this, change the number and letters separately:

0.5 × 5 × a × b \u003d 2,5ab

2,5ab \u003d 2,5ab

As a result of a small identical transformation, the left side of the equality has become equal to the right part of equality. So we have proven equality 0,5A × 5B \u003d 2,5ab Is identity.

Of identical transformations We learned to fold, subtract, multiply and divide the numbers, cut the fractions, bring similar terms, and simplify some expressions.

But this is not all identical transformations that exist in mathematics. Identical transformations are much more. In the future, we will be convinced more than once.

Tasks for self solutions:

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It is known that in mathematics it is not to do without simplifying expressions. This is necessary for the correct and quick solution A variety of tasks, as well as various kinds of equations. The simplification discussed implies a decrease in the number of actions necessary to achieve the goal. As a result, the calculation is alleviated noticeable, and time is significantly saved. But how to simplify the expression? This uses established mathematical relations, often referred to as formulas, or by laws that allow the expression much shorter, thus simplifying calculations.

It is no secret that today is not difficult to simplify the expression online. We give references to some of the most popular of them:

However, it is so possible to do with each expression. Therefore, we consider more traditional methods.

Taking a common divider

In the case when in one expression are present, possessing the same multipliers, you can find the amount of coefficients with them, and then multiply the multiplier for them. This operation is also called "Making a General Divider". Seriously using this method, sometimes you can significantly simplify the expression. Algebra After all, in general, in general, built on the grouping and regrouping of multipliers and divisors.

The simplest formulas of abbreviated multiplication

One consequence of the previously described method is the formulas of abbreviated multiplication. How to simplify the expressions with their help is much clearer to those who did not even remove these formulas by heart, but he knows that they are derived, that is, from where they come from, and accordingly, their mathematical nature. In principle, the previous statement retains its strength in all modern mathematics, starting from the first class and ending with the highest courses of mechanical and mathematical faculties. The difference of squares, the square of the difference and sum, the amount and difference of cubes - all these formulas are used everywhere in the elementary, as well as the highest mathematics in cases where it is necessary to simplify the expression to solve the tasks. Examples of such transformations can be easily found in any school textbook on algebra, or, which is even easier, on the expanses of the worldwide network.

Degree roots

Elementary mathematics, if you look at it in general, armed not so and in many ways, with which you can simplify the expression. The degrees and actions with them are usually managed by most students are relatively easy. Only, many modern schoolchildren and students have considerable difficulties when it is necessary to simplify the expression with roots. And it is completely unfounded. Because the mathematical nature of the roots are no different from the nature of the same degrees with which, as a rule, difficulties are much smaller. It is known that the square root from the number, variable or expression is nothing other than the same number, a variable or expression into a "one second", the cubic root is the same in the degree of "one third" and so on according to the correspondence.

Simplify expressions with fractions

Consider also a common example of how to simplify the expression with fractions. In cases where expressions are natural fractionsYou should allocate a common multiplier from the denominator and the numerator, and then cut the fraction on it. When unrocked with the same faults, elevated to degrees, it is necessary to monitor when they are summarized for the equality of degrees.

Simplify the simplest trigonometric expressions

Some mansion is a conversation about how to simplify the trigonometric expression. The widest section of trigonometry is perhaps the first stage in which the mathematics studying will have to face several abstract concepts, tasks and methods of their solution. Here there are their respective formulas, the first of which is the main trigonometric identity. Having a sufficient mathematical mindset, you can trace the systematic excretion from this identity of all major trigonometric identities and formulas, among which the formulas of the difference and the amount of arguments, double, triple arguments, formulas of bringing and many others. Of course, it is not worth forgeting here the very first methods, like the total multiplier, which is fully used along with new methods and formulas.

To summarize the results, provide the reader a few general tips:

• The polynomials should be laid on multipliers, that is, to represent them in the form of a product of a certain number of factors - single-wing and polynomials. If there is such an opportunity, you must bear the general factor for brackets.
• It is still better to learn all the formula for abbreviated multiplication without exception. They are not so much, but they are the basis of simplifying mathematical expressions. Do not also forget about the method of allocating full squares in three-stale, which is reverse action To one of the formulas of abbreviated multiplication.
• All fractions existing in expression should be reduced as often as possible. At the same time, do not forget that only multipliers are reduced. In the case when the denominator and the numerator of algebraic fractions are multiplied by the same number that differs from zero, the values \u200b\u200bof fractions do not change.
• In general, all expressions can be converted by actions or a chain. The first method is more preferable, because The results of intermediate actions are checked easier.
• Frequently often B. mathematical expressions Have to extract roots. It should be remembered that the roots of even degrees can be removed only from negative number or expressions, and the roots of odd degrees are completely from any expressions or numbers.

We hope our article will help you, further, understand the mathematical formulas and teach them to apply them in practice.

I. Expressions in which, along with letters, the numbers, marks of arithmetic action and brackets can be used, are called algebraic expressions.

Examples of algebraic expressions:

2m -n; 3. · (2a + b); 0.24X; 0,3A -B. · (4a + 2b); a 2 - 2ab;

Since the letter in algebraic expression can be replaced by some various numbers, then the letter is called variable, and itself algebraic expression - expression with a variable.

II. If in algebraic expression letters (variables), replace them with values \u200b\u200band perform these actions, then the resulting number is called an algebraic expression value.

Examples. Find the value of the expression:

1) A + 2B -C at a \u003d -2; b \u003d 10; C \u003d -3.5.

2) | x | + | Y | - | z | at x \u003d -8; y \u003d -5; z \u003d 6.

Decision.

1) A + 2B -C at a \u003d -2; b \u003d 10; C \u003d -3.5. Instead of variables, we substitute their values. We get:

— 2+ 2 · 10- (-3,5) = -2 + 20 +3,5 = 18 + 3,5 = 21,5.

2) | x | + | Y | - | z | at x \u003d -8; y \u003d -5; z \u003d 6. Substitute specified values. Remember that the negative number module is equal to the opposite number, and the module of a positive number is equal to the very number. We get:

|-8| + |-5| -|6| = 8 + 5 -6 = 7.

III. The values \u200b\u200bof the letter (variable), under which the algebraic expression makes sense, is called the allowable values \u200b\u200bof the letter (variable).

Examples. Under what values \u200b\u200bof the variable expression does not make sense?

Decision. We know that it is impossible to divide to zero, therefore, each of these expressions will not make sense in the value of the letter (variable), which draws the denomoter of the fraction in zero!

In Example 1) this value is a \u003d 0. Indeed, if instead and substitute 0, then you need to share the number 6 to 0, and this can not be done. Answer: Expression 1) does not make sense at a \u003d 0.

In Example 2) the denominator x - 4 \u003d 0 at x \u003d 4, therefore, this value x \u003d 4 and cannot be taken. Answer: Expression 2) does not make sense at x \u003d 4.

In Example 3) denominator x + 2 \u003d 0 at x \u003d -2. Answer: Expression 3) does not make sense at x \u003d -2.

In example 4) denominator 5 - | x | \u003d 0 with | x | \u003d 5. And since | 5 | \u003d 5 and | -5 | \u003d 5, then it is impossible to take x \u003d 5 and x \u003d -5. Answer: Expression 4) does not make sense at x \u003d -5 and at x \u003d 5.
IV. Two expressions are identically equal, if with any valid values \u200b\u200bof the variables, the corresponding values \u200b\u200bof these expressions are equal.

Example: 5 (a - b) and 5a - 5b are shadely equal, since equality 5 (a - b) \u003d 5a - 5b will be faithful at any values \u200b\u200bof A and b. Equality 5 (A - B) \u003d 5A - 5B There is a identity.

Identity - This is equality, just with all the permissible values \u200b\u200bof the variables included in it. Examples of identities already known to you are, for example, the properties of addition and multiplication, the distribution property.

The replacement of one expression to another, identically equal to it by the expression, is called identical conversion or simply by the transformation of the expression. The identity transformations of expressions with variables are based on the properties of the number of numbers.

Examples.

a) Convert the expression to identically equal, using the distribution property of multiplication:

1) 10 · (1.2x + 2.3,); 2) 1.5 · (A -2B + 4C); 3) A · (6m -2n + k).

Decision. Recall the distribution property (law) of multiplication:

(A + B) · C \u003d A · C + B · C (The distribution law of multiplication relative to addition: to multiply the amount of two numbers to the third number, you can multiply each component to this number and folded the results).
(A-B) · C \u003d A · C-B · C (Distribution law of multiplication relative to subtraction: To multiply the difference between two numbers to multiply by the third number, you can multiply by this number reduced and subtractable separately and from the first result of subtracting the second one).

1) 10 · (1.2x + 2,31) \u003d 10 · 1.2x + 10 · 2.3U \u003d 12x + 23W.

2) 1.5 · (A -2B + 4C) \u003d 1,5A -3B + 6C.

3) A · (6m -2n + k) \u003d 6am -2an + AK.

b) Convert the expression to identically equal, using the Moveless and Fashion Properties (laws) of addition:

4) x + 4.5 + 2x + 6.5; 5) (3a + 2,1) + 7.8; 6) 5.4C -3 -2.5 -2.3C.

Decision. Apply the laws (properties) of addition:

a + b \u003d b + a (Movement: the amount does not change from the rearrangement of the terms).
(A + B) + C \u003d A + (B + C) (Combining: To add a third number to the sum of the two components, you can add the second and third amount to the first number).

4) x + 4.5 + 2x + 6.5 \u003d (x + 2x) + (4.5 + 6.5) \u003d 3x + 11.

5) (3a + 2,1) + 7.8 \u003d 3A + (2.1 + 7.8) \u003d 3a + 9.9.

6) 6) 5.4C -3 -2.5 -2.3c \u003d (5.4C -2.3C) + (-3 -2.5) \u003d 3.1С -5.5.

in) Convert the expression to identically equal, using the Multiplication Multiplication: Multiplication:

7) 4 · H. · (-2,5); 8) -3,5 · 2ow · (-one); 9) 3A. · (-3) · 2c.

Decision. Apply the laws (properties) of multiplication:

a · b \u003d b · a (Movement: From the permutation of multipliers, the work does not change).
(A · b) · C \u003d A · (B · C) (Combining: To multiply the work of two numbers to the third number, you can multiply the first number to the work of the second and third).

7) 4 · H. · (-2,5) = -4 · 2,5 · x \u003d -10x.

8) -3,5 · 2ow · (-1) \u003d 7th.

9) 3A. · (-3) · 2C \u003d -18As.

If the algebraic expression is given in the form of a reduced fraction, then using the crushing rule, it can be simplified, i.e. Replace identically equal to a simpler expression.

Examples. Simplify using the reduction of fractions.

Decision. Reduce the fraction - this means dividing its numerator and denominator to the same number (expression), different from zero. Fraction 10) will reduce on 3b.; fraction 11) will reduce on but and fraction 12) will reduce on 7N.. We get:

Algebraic expressions are used to compile formulas.

The formula is an algebraic expression recorded in the form of equality and expressing the relationship between two or several variables. Example: Formula Formula you know s \u003d V · T (S is the path traveled, V is speed, t - time). Remember what other formulas you know.

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Expressions, transformation of expressions

Powerful expressions (expressions with degrees) and their conversion

In this article we will talk about transforming expressions with degrees. First we will focus on transformations that are performed with expressions of any species, including powerful expressions, such as disclosing brackets, bringing similar terms. And then we will analyze the transformation inherent in expressions with degrees: work with the basis and indicator of the degree, the use of the properties of degrees, etc.

Navigating page.

What are power expressions?

The term "powerful expressions" practically does not occur to school textbooks of mathematics, but it often appears in collections of tasks, especially designed to prepare for EGE and OGE, for example,. After analyzing the tasks in which any actions are required with power expressions, it becomes clear that under power expressions understand the expressions containing in their degree records. Therefore, it is possible to accept such a definition for yourself:

Definition.

Power expressions - These are expressions containing degrees.

Here examples of power expressions. Moreover, we will submit them according to how the development of views on degree with a natural indicator to the degree with the actual indicator occurs.

As you know, first the acquaintance with the degree of number with a natural figure, at this stage the first simplest power expressions of type 3 2, 7 5 +1, (2 + 1) 5, (-0,1) 4, 3 · A 2 appear -A + a 2, x 3-1, (a 2) 3, etc.

A little later, the degree of number with an integer is studied, which leads to the emergence of power expressions with whole negative degrees, like the following: 3 -2, , A -2 + 2 · b -3 + C 2.

In high school, returned to degrees again. There is a degree with a rational indicator, which entails the appearance of appropriate power expressions: , , etc. Finally, discusses degrees with irrational indicators and comprising their expressions: ,.

The case listed by power expressions is not limited to: the variable penetrates further in terms of the extent, and there are such expressions 2 x 2 +1 or . And after acquaintance, expressions with degrees and logarithms begin to meet, for example, x 2 · LGX -5 · X LGX.

So, we dealt with the question, which represents powerful expressions. We will continue to learn to convert them.

The main types of transformations of power expressions

With power expressions you can perform any of the main identity transformations of expressions. For example, you can reveal brackets, replace numerical expressions their values, bring similar components, etc. Naturally, it should be necessary to comply with the procedure for performing actions. We give examples.

Example.

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

Decision.

According to the procedure for performing actions, first perform actions in brackets. There, firstly, we replace the degree 4 2 of its value 16 (see if necessary), and secondly, we calculate the difference 16-12 \u003d 4. Have 2 3 · (4 2 -12) \u003d 2 3 · (16-12) \u003d 2 3 · 4.

In the resulting expression, we replace degree 2 3 of its value 8, after which we calculate the product 8 · 4 \u003d 32. This is the desired value.

So, 2 3 · (4 2 -12) \u003d 2 3 · (16-12) \u003d 2 3 · 4 \u003d 8 · 4 \u003d 32.

Answer:

2 3 · (4 2 -12) \u003d 32.

Example.

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

Decision.

It is obvious that this expression contains similar terms 3 · a 4 · b -7 and 2 · a 4 · b -7, and we can lead them :.

Answer:

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

Example.

Present an expression with degrees in the form of a work.

Decision.

Credit with the task allows the representation of the number 9 in the form of degree 3 2 and the subsequent use of the formula of the abbreviated multiplication. Square differences:

Answer:

There is also a number of identical transformations inherent in power expressions. Then we will discern them.

Work with the basis and indicator of the degree

There are extent, at the base and / or indicator of which are not just numbers or variables, but some expressions. As an example, give the record (2 + 0.3 · 7) 5-3.7 and (a · (a + 1) -a 2) 2 · (x + 1).

When working with similar expressions, it is possible as an expression at the base of the degree and the expression in the indicator to replace identically equal expression On the odd of its variables. In other words, we can separately convert the rootation of degree to us separately, and separately the indicator. It is clear that as a result of this transformation, an expression will be identically equal to the initial one.

Such transformations make it possible to simplify expressions with degrees or reach other purposes we need. For example, in the above-mentioned power expression (2 + 0.3 · 7) 5-3.7, it is possible to perform actions with numbers at the base and indicator, which will allow you to move to the degree of 4.1 1.3. And after the disclosures of the brackets and bringing similar terms at the base of the degree (A · (A + 1) -a 2) 2 · (x + 1) we get a power expression more simple view a 2 · (x + 1).

Use the properties of degrees

One of the main tools for transforming expressions with degrees is equality reflecting. Recall the main of them. For any positive numbers a and b and arbitrary valid numbers R and S are fair the following properties degrees:

• 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.

Note that with natural, integers, as well as the positive indicators of the degree of restriction on the number A and B may not be as strict. For example, for natural numbers m and n, the equality A m · a n \u003d a m + n is true not only for positive A, but also for negative, and for a \u003d 0.

At school, the focus on the transformation of power expressions is focused on the ability to select a suitable property and apply it correctly. At the same time, the bases of degrees are usually positive, which allows the use of the properties of degrees without restrictions. The same applies to the transformation of expressions containing variables in the bases of degrees - the area of \u200b\u200bpermissible values \u200b\u200bof variables is usually that the bases are taken only positive meaningsthat allows you to freely use the properties of degrees. In general, it is necessary to constantly wonder if it is possible to use any property of degrees in this case, because the inaccient use of properties can lead to a narrowing of OTZ and other troubles. In detail and on examples, these moments are disassembled in the article transformation of expressions using the properties of degrees. Here we will restrict ourselves to the consideration of several simple examples.

Example.

Prepare an expression A 2.5 · (A 2) -3: A -5.5 as a degree with a base a.

Decision.

First, the second factor (A 2) -3 is converting the exercise in the degree in the degree in the degree: (a 2) -3 \u003d a 2 · (-3) \u003d a -6. The initial power expression takes the form A 2.5 · A -6: A -5.5. Obviously, it remains to take advantage of the properties of multiplication and division of degrees with the same basis, we have
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 properties of degrees when converting power expressions are used both from left to right and right to left.

Example.

Find the value of a power expression.

Decision.

Equality (a · b) r \u003d a r · b R, applied to the right left, allows from the initial expression to move to the product and further. And when multiplying degrees with identical grounds Indicators fold: .

It was possible to perform the transformation of the initial expression and otherwise:

Answer:

.

Example.

The power expression A 1.5 -a 0.5 -6, enter a new variable T \u003d A 0.5.

Decision.

The degree A 1.5 can be represented as a 0.5 · 3 and on the database of the degree property to the degree (A R) S \u003d A R · S, applied to the right to left, convert it to the form (A 0.5) 3. In this way, a 1,5 -A 0.5 -6 \u003d (A 0.5) 3 -A 0.5 -6. Now it is easy to enter a new variable T \u003d A 0.5, we obtain T 3 -T-6.

Answer:

t 3 -T-6.

Transformation of fractions containing degrees

Powerful expressions may contain fractions with degrees or represent such fractions. Such fractions are fully applicable any of the main transformations of fractions that are inherent in fractions of any kind. That is, fractions that contain degrees can be reduced, lead to a new denominator, work separately with their numerator and separately with the denominator, etc. To illustrate the words, consider solutions of several examples.

Example.

Simplify power expression .

Decision.

This power expression is a fraction. We will work with its numerator and denominator. In the numerator, we will reveal the brackets and simplifies the expression obtained after this, using the properties of degrees, and in the denominator we will give similar terms:

And still change the sign of the denominator, placing minus before the fraction: .

Answer:

.

Bringing the degrees of fractions to a new denominator is carried out similarly to bringing rational fractions to a new denominator. At the same time, an additional factor is also located and multiplying the numerator and denominator of the fraction. Performing this action, it is worth remembering that bringing to a new denominator can lead to a narrowing of OTZ. To this not happen, it is necessary that the additional factor does not apply to zero at no matter what values \u200b\u200bof the variables from the odd variables for the initial expression.

Example.

Give fractions to a new denominator: a) to the denominator A, B) to the denominator.

Decision.

a) In this case, it is quite simple to imagine what an additional factor helps to achieve the desired result. This is a multiplier a 0.3, as a 0.7 · a 0.3 \u003d a 0.7 + 0.3 \u003d a. Note that on the area of \u200b\u200bpermissible values \u200b\u200bof the variable A (these are a plurality of all positive valid numbers) degree A 0.3 does not appeal to zero, therefore, we have the right to multiply the numerator and denominator of the specified fraction on this additional factor:

b) looking more closely to the denominator, it can be found that

And the multiplication of this expression on will give the amount of cubes and, that is,. And this is the new denominator to which we need to bring the original fraction.

So we found an additional factor. On the area of \u200b\u200bpermissible values \u200b\u200bof the variables x and y, the expression does not apply to zero, therefore, we can multiply the numerator and denominator of the fraction:

Answer:

but) b) .

There is nothing new in the reduction of fractions containing degrees, there is nothing new: the numerator and the denominator are represented as a number of multipliers, and the same multipliers of the numerator and the denominator are reduced.

Example.

Reduce the fraction: a) , b).

Decision.

a) firstly, the numerator and denominator can be reduced to numbers 30 and 45, which is equal to 15. Also, obviously, you can make a reduction on x 0.5 +1 and . That's what we have:

b) In this case, the same multipliers in the numerator and the denominator cannot be immediately visible. To get them, you will have to perform preliminary transformations. In this case, they are concluded in the expansion of the denominator for multipliers using the formula of the square difference:

Answer:

but)

b) .

Bringing fractions to a new denominator and the reduction of fractions is mainly used to perform action with fractions. Actions are performed according to the well-known rules. When adding (subtracting) fractions, they are given to a shared denominator, after which they are completed (subtracted) numerals, and the denominator remains the same. As a result, it turns out a fraction, the numerator of which is the product of numerals, and the denominator is a product of denominators. The division of the fraction is multiplication by fraction, inverse it.

Example.

Follow the steps .

Decision.

First, we perform the subtraction of fractions located in brackets. To do this, bring them to a common denominator who has , after which we subtract the numbers:

Now we multiply the fractions:

Obviously, it is possible to reduce the degree of x 1/2, after which we have .

You can still simplify the power expression in the denominator, using the formula of the square difference: .

Answer:

Example.

Simplify power expression .

Decision.

Obviously, this fraction can be reduced by (x 2.7 +1) 2, it gives a fraction . It is clear that you need to do something else with the degrees of ICA. To do this, we transform the resulting fraction into the work. This gives us the opportunity to take advantage of the property of degrees with the same grounds: . And in conclusion proceed from the last work To fraction.

Answer:

.

And I also add that it is possible and in many cases, it is desirable to transfer multiple degree rates from the numerator to a denominator or from the denominator to a numerator, changing the indicator sign. Such transformations often simplify further actions. For example, a power expression can be replaced by.

Transformation of expressions with roots and degrees

Often in expressions that require some transformations, along with degrees with fractional indicators there are roots. To convert a similar expression to listeningIn most cases, it is enough to go only to roots or only to degrees. But since it is more convenient to work with degrees, usually go from roots to degrees. However, it is advisable to exercise such a transition when the OTZ variables for the initial expression makes it possible to replace the roots by degrees without having to turn to the module or split OTZ to several gaps (we disassembled in detail the transition from the roots to the degrees and back after exploring the degree with a rational indicator The degree with the irrational indicator is introduced, which allows you to talk about the degree with an arbitrary real indicator. At this stage, the school begins to study exponential function which is analyzically defined by the degree in which the number is located, and in the indicator - the variable. So we are confronted with the powerful expressions containing the number in the foundation of the degree, and in the indicator - expressions with variables, and naturally there is a need to perform transformations of such expressions.

It should be said that the transformation of the expressions of the specified species usually has to be performed when solving indicatory equations and indicative inequalities And these transformations are quite simple. In the overwhelming number of cases, they are based on the degree properties and are aimed for the most part to enter a new variable in the future. Demonstrate them will allow the equation 5 2 · x + 1 -3 · 5 x · 7 x -14 · 7 2 · x-1 \u003d 0.

Firstly, the degrees in the indicators of which there is a sum of some variable (or expressions with variables) and the numbers are replaced by the works. This applies to the first and last term expressions from the left side:
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.

Further, the division of both parts of equality is performed on the expression 7 2 · x, which only positive values \u200b\u200btake on the source equation to the source equation (this is the standard reception of solving equations of this type, it is not about him now, so focus on subsequent transformations of expressions with degrees ):

Now the fractions are reduced with degrees, which gives .

Finally, the ratio of degrees with the same indicators is replaced by degrees of relations, which leads to the equation That is equivalent . Transformations made allow you to enter a new variable, which reduces the solution of the original indicative equation to solve the square equation

I. V. Boykov, L. D. Romanova Collection of tasks for preparation for the exam. Part 1. Penza 2003.