Simplify fractional expressions. How to simplify the mathematical expression

Often, the tasks require a simplified answer. Although simplified, and unprofitable answers are faithful, the teacher can reduce your assessment if you do not simplify the answer. Moreover, with a simplified mathematical expression, it is much easier to work. Therefore, it is very important to learn to simplify expressions.

Steps

The right procedure for performing mathematical operations

1. Remember the correct procedure for performing mathematical operations. When simplifying a mathematical expression, it is necessary to observe a certain procedure, since some mathematical operations have priority over others and must be made first (in fact, non-compliance with the correct procedure for performing operations will lead you to an incorrect result). Remember next order Performing mathematical operations: expression in brackets, exercise into a degree, multiplication, division, addition, subtraction.

   • Please note that the knowledge of the correct order of operation will allow you to simplify most of the simplest expressions, but to simplify the polynomial (expressions with the variable) you need to know special techniques (see the next section).

2. Start with solutions to expressions in brackets. In mathematics, the brackets indicate that the expression concluded in them should be fulfilled in the first place. Therefore, when simplifying any mathematical expression, start with the solution of expression enclosed in the bracket (no matter what operations need to be performed inside the brackets). But remember that working with an expression concluded in brackets should be followed by the procedure for conducting operations, that is, members in brackets are first multiplied, divided, add up, deducted and so on.

   • For example, we simplify expression 2x + 4 (5 + 2) + 3 2 - (3 + 4/2). Here, let's start with expressions in brackets: 5 + 2 \u003d 7 and 3 + 4/2 \u003d 3 + 2 \u003d 5.
     • The expression in the second pair of brackets is simplified to 5, because you first need to divide 4/2 (according to the correct procedure for performing operations). If you do not observe this order, then you will receive the wrong answer: 3 + 4 \u003d 7 and 7 ÷ 2 \u003d 7/2.
   • If there are one more pair of brackets in brackets, start simplifying from solving an expression in internal brackets, and then proceed to the solution of expression in external brackets.

3. Early degree. Deciding expressions in brackets, go to the exercise to the extent (remember that the degree is an indicator of the degree and the foundation of the degree). Take the appropriate expression (or number) to the degree and substitute the result in the expression given to you.

   • In our example, the only expression (number) is to the degree 3 2: 3 2 \u003d 9. In this image, instead of 3 2, substitute 9 and you will receive: 2x + 4 (7) + 9 - 5.

4. Multiply. Remember that the multiplication operation can be denoted by the following symbols: "X", "∙" or "*". But if between the number and variable (for example, 2x) or between the number and number in brackets (for example, 4 (7)) there are no characters, then it is also a multiplication operation.

   • In our example, there are two multiplication operations: 2x (two multiply to the variable "x") and 4 (7) (multiply seven). We do not know the meaning x, so the expression 2x will leave as it is. 4 (7) \u003d 4 x 7 \u003d 28. Now you can rewrite the expression given to you: 2x + 28 + 9 - 5.

5. Divide. Remember that the division operation may be denoted by the following symbols: "/", "÷" or "-" (you can meet the last character in the frauds). For example, 3/4 is three divided by four.

   • In our example, the division operation no longer, since you have already divided 4 to 2 (4/2) when solving an expression in brackets. Therefore, you can go to the next step. Remember that in most expressions there are no mathematical operations at once (only some of them).

6. Fold. With the addition of members of the expression, you can start from the most extreme (left) member, or you can first fold the members of the expression that are easy to fold. For example, in the expression 49 + 29 + 51 +71, it is first easier to add 49 + 51 \u003d 100, then 29 + 71 \u003d 100 and, finally, 100 + 100 \u003d 200. It is much more difficult to fold this: 49 + 29 \u003d 78; 78 + 51 \u003d 129; 129 + 71 \u003d 200.

   • In our example 2x + 28 + 9 + 5 there are two additions of addition. Let's start with the most extreme member: 2x + 28; You can not fold 2x and 28, because you do not know the values \u200b\u200bof the variable "x". Therefore, fold 28 + 9 \u003d 37. Now the expression can be rewritten so: 2x + 37 - 5.

7. Remove. This is the last operation in the right order performing mathematical operations. At this stage, you can also add negative numbers or do it at the stage of addition of members - this will not affect the final result.

   • In our example, 2x + 37 - 5 there is only one subtraction operation: 37 - 5 \u003d 32.

8. At this stage, having done all mathematical operations, you should get a simplified expression. But if the expression given to you contains one or more variables, remember that the member with the variable will remain as it is. The solution (and not simplification) of the expression with the variable implies finding the value of this variable. Sometimes an expressions with a variable can be simplified using special methods (see the next section).

   • In our example, the final answer: 2x + 32. You will not be able to fold two members until you know the value of the variable "x". Learning the importance of the variable, you can easily simplify this twist.

   Simplify complex expressions

   1. Addition of such members. Remember that it is possible to deduct and fold only such members, that is, members with the same variable and the same indicator. For example, you can add 7x and 5x, but it is impossible to fold 7x and 5x 2 (as the indicators of the degree of different).

      • This rule applies to members with several variables. For example, you can fold 2XY 2 and -3XY 2, but you can not fold 2XY 2 and -3X 2 Y or 2XY 2 and -3Y 2.
      • Consider an example: x 2 + 3x + 6 - 8x. Here, 3x and 8x are similar members, so they can be folded. The simplified expression looks like this: x 2 - 5x + 6.

   2. Simplify the numeric fraction. In such a fraction and in the numerator, and in the denominator there are numbers (without variable). The numerical fraction is simplified in several ways. First, just divide the denominator to the numerator. Secondly, spread the numerator and denominator for multipliers and reduce the same multipliers (since when dividing the number on itself, you will receive 1). In other words, if the numerator is, and the denominator has the same factor, it can be discarded and getting a simplified fraction.

      • For example, consider the fraction 36/60. With the help of a calculator, divide 36 to 60 and get 0.6. But you can simplify this fraction and differently, decomposing the numerator and denominator for multipliers: 36/60 \u003d (6x6) / (6x10) \u003d (6/6) * (6/10). Since 6/6 \u003d 1, then simplified fraction: 1 x 6/10 \u003d 6/10. But this fraction can also be simplified: 6/10 \u003d (2x3) / (2 * 5) \u003d (2/2) * (3/5) \u003d 3/5.

   3. If the fraction contains a variable, you can reduce the same multipliers with a variable. Spread and numerator, and denominator for multipliers and reduce the same multipliers, even if they contain a variable (remember that here the same multipliers may contain or contain a variable).

      • Consider an example: (3x 2 + 3x) / (- 3x 2 + 15x). This expression can be rewritten (decompose on multipliers) in the form: (x + 1) (3x) / (3x) (5 - x). Since a member of 3x is both in the numerator, and in the denominator, it can be reduced, and you will get a simplified expression: (x + 1) / (5 - x). Consider another example: (2x 2 + 4x + 6) / 2 \u003d (2 (x 2 + 2x + 3)) / 2 \u003d x 2 + 2x + 3.
      • Please note that you cannot reduce any members - only the same multipliers are reduced, which are present both in the numerator and in the denominator. For example, in expression (x (x + 2)) / x, the variable (multiplier) "X" is both in the numerator, and in the denominator, so "x" can be reduced and obtain a simplified expression: (x + 2) / 1 \u003d x + 2. However, in the expression (x + 2) / x variable "X" cannot be reduced (since the "X" numerator is not a multiplier).

   4. Open parenthesis. To do this, multiply a member behind the bracket for each member in brackets. Sometimes it helps to simplify complex expression. This applies to both members that are simple numbers and to members that contain a variable.

      • For example, 3 (x 2 + 8) \u003d 3x 2 + 24, and 3x (x 2 + 8) \u003d 3x 3 + 24x.
      • Please note that in fractional expressions, the brackets are not necessary, if in the numerator, and in the denominator there is the same multiplier. For example, in the expression (3 (x 2 + 8)) / 3x brackets, it is not necessary to disclose, since here you can reduce the multiplier 3 and obtain a simplified expression (x 2 + 8) / x. With this expression it is easier to work; If you were revealed brackets, you would get the following complex expression: (3x 3 + 24x) / 3x.

   5. Spread on multipliers of polynomials. With this method, you can simplify some expressions and polynomials. Decomposition of multipliers is an operation opposite to disclosure of brackets, that is, the expression is recorded as a product of two expressions, each of which is enclosed in brackets. In some cases, the expansion of multipliers reduces the same expression. In special cases (as a rule, with square equations) Decomposition of multipliers will allow you to solve the equation.

      • Consider the expression X 2 - 5X + 6. It is decomposed on the multipliers: (x - 3) (x - 2). Thus, if, for example, expression (x 2 - 5x + 6) / (2 (x - 2)), then you can rewrite it in the form (x - 3) (x - 2) / (2 (x - 2)), reduce expression (x - 2) and obtain a simplified expression (x - 3) / 2.
      • The expansion of polynomials on factors is used to solve (finding roots) of equations (equation is a polynomial equivalent to 0). For example, consider the equation x 2 - 5x + 6 \u003d 0. Decuting it to multipliers, you will get (x - 3) (x - 2) \u003d 0. Since any expression multiplied by 0, equal to 0, then we can write so : x - 3 \u003d 0 and x - 2 \u003d 0. Thus, x \u003d 3 and x \u003d 2, that is, you found two roots of the equations given to you.

Among the various expressions, which are considered in algebra, the amount of homorals occupy an important place. We give examples of such expressions:
\\ (5a ^ 4 - 2a ^ 3 + 0,3a ^ 2 - 4,6A + 8 \\)
\\ (xy ^ 3 - 5x ^ 2y + 9x ^ 3 - 7Y ^ 2 + 6x + 5y - 2 \\)

The amount of homorals is called polynomial. The components in the polynomial are called members of the polynomial. We are also unintently refer to the polynomials, counting is unintently by a polynomial consisting of one member.

For example, polynomial
\\ (8B ^ 5 - 2B \\ Cdot 7b ^ 4 + 3b ^ 2 - 8b + 0.25b \\ Cdot (-12) B + 16 \\)
You can simplify.

Imagine all the terms in the form of homorals standard view:
\\ (8B ^ 5 - 2B \\ CDOT 7B ^ 4 + 3B ^ 2 - 8B + 0.25B \\ CDOT (-12) B + 16 \u003d \\)
\\ (\u003d 8b ^ 5 - 14b ^ 5 + 3b ^ 2 -8b -3b ^ 2 + 16 \\)

We give such members in the resulting polynomial:
\\ (8b ^ 5 -14b ^ 5 + 3b ^ 2 -8b -3b ^ 2 + 16 \u003d -6b ^ 5 -8b + 16 \\)
It turned out a polynomial, all members of which are one-sided species, and there are no similar among them. Such polynomials are called polynomials of standard species.

Per the degree of polynomial The standard species take the largest of the degrees of its members. Thus, bicked \\ (12a ^ 2b - 7b \\) has a third degree, and three stages \\ (2b ^ 2 -7b + 6 \\) - the second.

Typically, members of the polynomials of a standard form containing one variable are placed in the order of decrease in its degree. For example:
\\ (5x - 18x ^ 3 + 1 + x ^ 5 \u003d x ^ 5 - 18x ^ 3 + 5x + 1 \\)

The sum of several polynomials can be converted (simplify) into a polynomial of a standard species.

Sometimes members of the polynomial need to be divided into groups by entering into each group in brackets. Since conclusion in brackets is a transformation, reverse disclosure of brackets, it is easy to formulate rules for disclosing brackets:

If the "+" sign is set in front of the brackets, the members enclosed in brackets are recorded with the same signs.

If the "-" sign is installed in front of the brackets, the members concluded in the brackets are recorded with opposite signs.

Transformation (simplification) of works of single-wing and polynomial

Using the distribution properties of multiplication, you can convert (simplify) into a polynomial, the product is unoblared and polynomial. For example:
\\ (9a ^ 2b (7a ^ 2 - 5ab - 4b ^ 2) \u003d \\)
\\ (\u003d 9a ^ 2b \\ Cdot 7a ^ 2 + 9a ^ 2B \\ CDOT (-5AB) + 9A ^ 2B \\ CDOT (-4B ^ 2) \u003d \\)
\\ (\u003d 63a ^ 4b - 45a ^ 3b ^ 2 - 36A \u200b\u200b^ 2b ^ 3 \\)

The work is unobed and the polynomial is identically equal to the amount of works of this single and each of the members of the polynomial.

This result is usually formulated as a rule.

To multiply unripe of a polynomial, you need to multiply this one is unknown for each of the members of the polynomial.

We have repeatedly used this rule for multiplication by the amount.

The product of polynomials. Transformation (simplification) works of two polynomials

In general, the product of two polynomials is identically equal to the amount of the work of each member of one polynomial and each member of the other.

Usually enjoy the following rule.

To multiply the polynomial to the polynomial, each member of one polynomial is multiplied by each member of the other and folded the obtained works.

Formulas of abbreviated multiplication. Squares of the amount, difference and difference of squares

With some expressions in algebraic transformations, it is necessary to deal more often than with others. Perhaps the most common expressions \\ ((a + b) ^ 2, \\; (a - b) ^ 2 \\) and \\ (a ^ 2 - b ^ 2 \\), i.e., the sum of the sum, the square of the difference and Square differences. You noticed that the names of the specified expressions are not over, so, for example, \\ ((a + b) ^ 2 \\) is, of course, not just the square of the amount, and the square of the sum A and B. However, the square of the amount A and B is not so often, as a rule, instead of letters a and b, it turns out to be different, sometimes quite complex expressions.

Expressions \\ ((a + b) ^ 2, \\; (a - b) ^ 2 \\) It is not difficult to convert (simplify) into polynomials of a standard species, in fact, you have already met with such a task when multiplying polynomials:
\\ ((a + b) ^ 2 \u003d (a + b) (a + b) \u003d a ^ 2 + AB + Ba + B ^ 2 \u003d \\)
\\ (\u003d a ^ 2 + 2ab + b ^ 2 \\)

The obtained identities are useful to remember and apply without intermediate calculations. A brief verbal wording helps this.

\\ ((a + b) ^ 2 \u003d a ^ 2 + b ^ 2 + 2ab \\) - the sum of the sum is equal to the sum of the squares and the doubled work.

\\ ((a - b) ^ 2 \u003d a ^ 2 + b ^ 2 - 2ab \\) - the square of the difference is equal to the sum of the squares without a double product.

\\ (a ^ 2 - b ^ 2 \u003d (a - b) (a + b) \\) - the difference of squares is equal to the product of the difference in the amount.

These three identities allow in transformations to replace their left parts with the right and back - right parts left. The most difficult at the same time - see the appropriate expressions and understand how variables A and B are replaced. Consider several examples of using the formulas of abbreviated multiplication.

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 module negative number It 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. Identical transformations Expansions with variables are performed based on the properties of actions above the 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 is reduced and subtractable separately and from the first result of the subtraction of 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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Using any language, you can express the same information. different words and turns. Not exception and mathematical language. But the same expression can be recorded equivalently in different ways. And in some situations, one of the records is simpler. We will talk about simplifying expressions in this lesson.

People communicate on different languages. For us, an important comparison is a pair of "Russian language - Mathematical language". The same information can be reported in different languages. But, in addition, it can also pronounce in one language in different ways.

For example: "Petya is friends with Vasya", "Vasya is friends with Petya", "Pete with Vay Friends." Said differently, but the same thing. For any of these phrases, we would understand what we are talking about.

Let's look at this phrase: "Petya's boy and boy Vasya are friends." We understood what this is speech. Nevertheless, we do not like how this phrase sounds. Can we simplify it, say the same, but easier? "Boy and boy" - you can say once again: "Petya and Vasya boys are friends."

"Boys" ... Isn't the names that they are not girls. We remove the "boys": "Petya and Vasya are friends." And the word "Friends" can be replaced with "Friends": "Peter and Vasya - Friends." As a result, the first, long ugly phrase was replaced by an equivalent statement, which is easier to say and easier to understand. We simplified this phrase. Simplify - it means to say easier, but not to lose, do not distort the meaning.

In a mathematical language, approximately the same thing happens. One thing can be said to write differently. What does it mean to simplify expression? This means that there are many equivalent expressions for the initial expression, that is, those that mean the same thing. And from all this set, we must choose the simplest, in our opinion, or the most suitable for our future goals.

For example, consider a numerical expression. It will be equivalent.

It will also be equivalent to the first two: .

It turns out that we have simplified our expressions and found the most brief equivalent expression.

For numerical expressions You always need to perform all actions and get an equivalent expression in the form of one number.

Consider an example of an alphabetic expression. . Obviously, it will be simpler.

When simplifying alphabetic expressions, you must perform all actions that are possible.

Do you always need to simplify expression? No, sometimes it will be more convenient for us equivalent, but a longer recording.

Example: From the number you need to take away the number.

It is possible to calculate, but if the first number was represented by its equivalent record:, then the calculations would be instantaneous :.

That is, a simplified expression is not always profitable for further computing.

Nevertheless, very often we face a task that it sounds "to simplify the expression".

Simplify the expression :.

Decision

1) Perform actions in the first and second brackets :.

2) Calculate the works: .

Obviously, the last expression is a simpler view than the initial one. We simplified it.

In order to simplify the expression, it must be replaced with an equivalent (equal).

To determine the equivalent expression, it is necessary:

1) perform all possible actions

2) Use the properties of addition, subtraction, multiplication and divisions to simplify calculations.

Properties of addition and subtraction:

1. Move the property of addition: the amount does not change from the rearrangement of the terms.

2. The combination property of the addition: to add a third number to the sum of two numbers, you can add the sum of the second and third number to the first number.

3. The property of subtraction of the amount from among: To subtract the amount from the number, you can deduct each term separately.

Properties of multiplication and division

1. Movement property of multiplication: the product does not change from the permutation of multipliers.

2. Fashionable property: To multiply the number on the work of two numbers, you can first multiply it to the first factor, and then the resulting work is multiplied by the second factor.

3. The distribution property of multiplication: to multiply the number to the amount, you need to multiply it to each alone separately.

Let's see how we actually make calculations in the mind.

Calculate:

Decision

1) imagine how

2) Imagine the first factor as the sum of the discharge terms and perform multiplication:

3) You can imagine how to perform multiplication:

4) Replace the first factor of the equivalent amount:

Distribution law can be used in reverse side: .

Perform actions:

1) 2)

Decision

1) For convenience, you can use the distributional law, just to use it in the opposite direction - to make a general factor for brackets.

2) I will bring a general multiplier for brackets.

It is necessary to buy linoleum in the kitchen and an entrance hall. Square kitchen -, hallway -. There are three types of linoleums: software, and rubles for. How much will each of three species Linoleum? (Fig. 1)

Fig. 1. Illustration to the condition of the problem

Decision

Method 1. You can individually find how much money will need to buy a linoleum into the kitchen, and then add to the hallway and the obtained works.

Some algebraic examples are able to bring horror on schoolchildren. Long expressions are not only frightened, but also make it difficult to calculate. Trying to make it possible to understand what and for what should be confused for a short time. It is for this reason that mathematics are always trying to simplify the "terrible" task as much as possible and only then proceed to his decision. Oddly enough, such a trick significantly speeds up the work process.

Simplification is one of the fundamental moments in the algebra. If in simple tasks without it you can still do, then more difficult for calculating examples may be "not on the teeth". This is where these skills will come in handy! Especially since complex mathematical knowledge is not required: it will be enough just to remember and learn to apply several basic techniques in practice and formulas.

Regardless of the complexity of calculations in solving any expression, it is important observe the procedure for performing operations with numbers:

1. brackets;
2. exercise;
3. multiplication;
4. division;
5. addition;
6. subtraction.

The last two points can be safely changed in places and it will not affect the result. But folding two adjacent numbers, when a multiplication sign is categorically impossible next to one of them! The answer is if it turns out, then incorrect. Therefore, you need to remember the sequence.

Application of similar

Such elements include numbers from a variable of one order or the same degree. There are so-called free members who do not have alphabetic designation unknown.

The essence is that in the absence of brackets you can simplify the expression, folding or subtracting similar.

Several visual examples:

• 8x 2 and 3x 2 - both numbers have the same second order variable, so they are similar and when addition is simplified to (8 + 3) x 2 \u003d 11x 2, while at subtraction it turns out (8-3) x 2 \u003d 5x 2;
• 4x 3 and 6x - and here "X" has a different degree;
• 2Y 7 and 33X 7 - contain various variables, therefore, as in the previous case, do not relate to the like.

Multiplier decomposition

This little mathematical trick, if you learn how to use it correctly, in the future it will not be able to cope with the tricky task. Yes, and understand how "System" works, it is easy: decomposition refer to the product of several elements, the calculation of which gives the initial value. Thus, 20 can be represented as 20 × 1, 2 × 10, 5 × 4, 2 × 5 × 2 or in another way.

On a note: Multiplers always coincide with divisors. So look for a working "pair" to decomposition, it is necessary among the numbers to which the initial is divided without a residue.

You can make such an operation as with free members and with numbers with a variable. The main thing is not to lose the latter during computing - even after decomposition, the unknown cannot take and "go nowhere." It remains with one of the multipliers:

• 15x \u003d 3 (5x);
• 60U 2 \u003d (15Y 2) 4.

Simple numbers that can be divided only on themselves or 1, never laid out - this makes no sense.

Main ways of simplification

The first, for which the look is clinging:

• the presence of brackets;
• fractions;
• roots.

Algebraic examples B. school Program Often are compiled taking into account the fact that they can be beautifully simplified.

Calculations in brackets

Carefully follow the sign facing brackets! Multiplication or division applies to each element inside, and minus - changes the existing signs "+" or "-" to the opposite.

Brackets are calculated according to the rules either by abbreviated multiplication formulas, after which the likes are given.

Reducing fractions

Shrink Also easy. They themselves "willingly run away", it is worth performing operations with bringing such members. But you can simplify an example before: pay attention to the numerator and denominator. They often contain explicit or hidden elements that can be mutually reduced. True, if in the first case you only need to strike unnecessary, in the second you will have to think, leading part of the expression to the form to simplify. Methods used:

• search and submission for the parents of the largest common divider in the numerator and denominator;
• division of everyone top element On the denominator.

When an expression or part of it is under the rootThe primary task of simplification is almost similar to the case with fractions. It is necessary to look for ways to get rid of it completely or, if it is impossible, to reduce the computing sign as much as possible. For example, to unobtrusive √ (3) or √ (7).

True way Simplify the past expression - try to decompose it for multipliersThe part of which is endowed beyond the sign. Visual example: √ (90) \u003d √ (9 × 10) \u003d √ (9) × √ (10) \u003d 3√ (10).

Other little tricks and nuances:

• this simplification operation can be carried out with fractions, making it for a sign as entirely and separately a numerator or denominator;
• lay out and end the amount of the amount or difference from the root;
• when working with variables, be sure to take into account its degree, it must be equal or a multiple root for the possibility of making: √ (x 2 y) \u003d x√ (y), √ (x 3) \u003d √ (x 2 × x) \u003d x√ ( x);
• sometimes it is allowed to get rid of the baked variable by erection in a fractional degree: √ (y 3) \u003d y 3/2.

Simplifying power expression

If in the case of simple calculations for minus or plus examples are simplified due to bringing the likes, how to be when multiplying or dividing variables with different degrees? They can be easily simplified, remembering two main points:

1. If there is a sign of multiplication between variables - degrees.
2. When they are divided into each other - it is deducted from the degree of the numerator.

The only condition for such a simplification - same base At both members. Examples for clarity:

• 5x 2 × 4x 7 + (y 13 / y 11) \u003d (5 × 4) x 2 + 7 + y 13- 11 \u003d 20x 9 + y 2;
• 2z 3 + z × z 2 - (3 × z 8 / z 5) \u003d 2z 3 + z 1 + 2 - (3 × z 8-5) \u003d 2z 3 + z 3 -3z 3 \u003d 3z 3 -3z 3 \u003d 0.

We note that operations with numerical valuesfacing variables occur by ordinary mathematical rules. And if you look at, it becomes clear that the power elements of the expression "work" is similar:

• the erection of a member in the degree refers to the multiplication of it on itself a certain number of times, i.e. x 2 \u003d x × x;
• the division is similar: if you decompose the degree of the numerator and the denominator, then some of the variables will decrease, while the remaining "collected", which is equivalent to subtraction.

As in any case, when simplifying algebraic expressions, not only knowledge of the basics is necessary, but also practice. Already in several occupations, examples, sometime seeming difficult, will be reduced without much difficulty, turning into short and easily solved.

Video

This video will help you figure out and remember how expressions are simplified.

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