How to simplify an expression examples. How to simplify algebraic expressions

§ 1 The concept of simplifying a literal expression

In this lesson, we will get acquainted with the concept of “similar terms” and, using examples, we will learn how to perform the reduction of similar terms, thus simplifying literal expressions.

Let’s find out the meaning of the concept “simplification”. The word “simplification” is derived from the word “simplify”. To simplify means to make simple, simpler. Therefore, to simplify a literal expression is to make it shorter, with minimum quantity actions.

Consider the expression 9x + 4x. This is a literal expression that is a sum. The terms here are presented as products of a number and a letter. The numerical factor of such terms is called a coefficient. In this expression, the coefficients will be the numbers 9 and 4. Please note that the factor represented by the letter is the same in both terms of this sum.

Let us recall the distributive law of multiplication:

To multiply a sum by a number, you can multiply each term by that number and add the resulting products.

IN general view written as follows: (a + b) ∙ c = ac + bc.

This law is true in both directions ac + bc = (a + b) ∙ c

Let's apply it to our literal expression: the sum of the products of 9x and 4x is equal to the product whose first factor is equal to the sum 9 and 4, the second factor is x.

9 + 4 = 13, that's 13x.

9x + 4 x = (9 + 4)x = 13x.

Instead of three actions in the expression, there is only one action left - multiplication. This means that we have made our literal expression simpler, i.e. simplified it.

§ 2 Reduction of similar terms

The terms 9x and 4x differ only in their coefficients - such terms are called similar. The letter part of similar terms is the same. Similar terms also include numbers and equal terms.

For example, in the expression 9a + 12 - 15 similar terms will be the numbers 12 and -15, and in the sum of the product of 12 and 6a, the number 14 and the product of 12 and 6a (12 ∙ 6a + 14 + 12 ∙ 6a) the equal terms represented by the product of 12 and 6a.

It is important to note that terms whose coefficients are equal, but whose letter factors are different, are not similar, although it is sometimes useful to apply the distributive law of multiplication to them, for example, the sum of the products 5x and 5y is equal to the product of the number 5 and the sum of x and y

5x + 5y = 5(x + y).

Let's simplify the expression -9a + 15a - 4 + 10.

Similar terms in in this case are the terms -9a and 15a, since they differ only in their coefficients. Their letter multiplier is the same, and the terms -4 and 10 are also similar, since they are numbers. Add up similar terms:

9a + 15a - 4 + 10

9a + 15a = 6a;

We get: 6a + 6.

By simplifying the expression, we found the sums of similar terms; in mathematics this is called reduction of similar terms.

If adding such terms is difficult, you can come up with words for them and add objects.

For example, consider the expression:

For each letter we take our own object: b-apple, c-pear, then we get: 2 apples minus 5 pears plus 8 pears.

Can we subtract pears from apples? Of course not. But we can add 8 pears to minus 5 pears.

Let us present similar terms -5 pears + 8 pears. Similar terms have the same letter part, so when bringing similar terms it is enough to add the coefficients and add the letter part to the result:

(-5 + 8) pears - you get 3 pears.

Returning to our literal expression, we have -5 s + 8 s = 3 s. Thus, after bringing similar terms, we obtain the expression 2b + 3c.

So, in this lesson you became acquainted with the concept of “similar terms” and learned how to simplify letter expressions by reducing similar terms.

List of used literature:

1. Mathematics. 6th grade: lesson plans to the textbook I.I. Zubareva, A.G. Mordkovich // author-compiler L.A. Topilina. Mnemosyne 2009.
2. Mathematics. 6th grade: textbook for students educational institutions. I.I. Zubareva, A.G. Mordkovich. - M.: Mnemosyne, 2013.
3. Mathematics. 6th grade: textbook for general education institutions/G.V. Dorofeev, I.F. Sharygin, S.B. Suvorov and others/edited by G.V. Dorofeeva, I.F. Sharygina; Russian Academy of Sciences, Russian Academy of Education. M.: “Enlightenment”, 2010.
4. Mathematics. 6th grade: study for general educational institutions/N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwartzburd. – M.: Mnemosyne, 2013.
5. Mathematics. 6th grade: textbook/G.K. Muravin, O.V. Muravina. – M.: Bustard, 2014.

Images used:

Often tasks require a simplified answer. Although both simplified and unsimplified answers are correct, your instructor may lower your grade if you do not simplify your answer. Moreover, the simplified mathematical expression is much easier to work with. Therefore, it is very important to learn to simplify expressions.

Steps

Correct order of mathematical operations

1. Remember the correct order of execution mathematical operations. When simplifying a mathematical expression, it is necessary to observe certain order actions, since some mathematical operations take precedence over others and must be done first (in fact, not following the correct order of doing the operations will lead you to the wrong result). Remember the following order of mathematical operations: expression in parentheses, exponentiation, multiplication, division, addition, subtraction.

   • Note that knowing the correct order of operations will allow you to simplify most simple expressions, but to simplify a polynomial (an expression with a variable) you need to know special tricks (see the next section).

2. Start by solving the expression in parentheses. In mathematics, parentheses indicate that the expression within them must be evaluated first. Therefore, when simplifying any mathematical expression, start by solving the expression enclosed in parentheses (it does not matter what operations you need to perform inside the parentheses). But remember that when working with an expression enclosed in brackets, you must follow the order of operations, that is, the terms in brackets are first multiplied, divided, added, subtracted, and so on.

   • For example, let's simplify the expression 2x + 4(5 + 2) + 3 2 - (3 + 4/2). Here we start with the expressions in brackets: 5 + 2 = 7 and 3 + 4/2 = 3 + 2 =5.
     • The expression in the second pair of parentheses simplifies to 5 because 4/2 must be divided first (according to the correct order of operations). If you don't follow this order, you will get the wrong answer: 3 + 4 = 7 and 7 ÷ 2 = 7/2.
   • If there is another pair of parentheses in the parentheses, start simplifying by solving the expression in the inner parentheses and then move on to solving the expression in the outer parentheses.

3. Exponentiate. Having solved the expressions in parentheses, move on to exponentiation (remember that a power has an exponent and a base). Raise the corresponding expression (or number) to a power and substitute the result into the expression given to you.

   • In our example, the only expression (number) to the power is 3 2: 3 2 = 9. In the expression given to you, replace 3 2 with 9 and you will get: 2x + 4(7) + 9 - 5.

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

   • In our example, there are two multiplication operations: 2x (two multiplied by the variable “x”) and 4(7) (four multiplied by seven). We don't know the value of x, so we'll leave the expression 2x as is. 4(7) = 4 x 7 = 28. Now you can rewrite the expression given to you as follows: 2x + 28 + 9 - 5.

5. Divide. Remember that the division operation can be represented by the following symbols: “/”, “÷” or “–” (you can see the last character in fractions). For example, 3/4 is three divided by four.

   • In our example, there is no longer a division operation, since you already divided 4 by 2 (4/2) when solving the expression in parentheses. So you can go to next step. Remember that most expressions do not contain all the mathematical operations (only some of them).

6. Fold. When adding terms of an expression, you can start with the term on the farthest (to the left), or you can add the terms that add easily first. For example, in the expression 49 + 29 + 51 +71, it is first easier to add 49 + 51 = 100, then 29 + 71 = 100 and finally 100 + 100 = 200. It is much more difficult to add like this: 49 + 29 = 78; 78 + 51 = 129; 129 + 71 = 200.

   • In our example 2x + 28 + 9 + 5 there are two addition operations. Let's start with the outermost (left) term: 2x + 28; you can't add 2x and 28 because you don't know the value of the variable "x". Therefore, add 28 + 9 = 37. Now the expression can be rewritten as follows: 2x + 37 - 5.

7. Subtract. This last operation V in the right order performing mathematical operations. At this stage you can also add negative numbers or do it at the stage of adding members - this will not affect the final result in any way.

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

8. At this stage, after doing all the mathematical operations, you should get a simplified expression. But if the expression given to you contains one or more variables, then remember that the term with the variable will remain as it is. Solving (not simplifying) an expression with a variable involves finding the value of that variable. Sometimes variable expressions can be simplified using special methods(see next section).

   • In our example, the final answer is 2x + 32. You cannot add the two terms until you know the value of the variable "x". Once you know the value of the variable, you can easily simplify this binomial.

   Simplifying complex expressions

   1. Addition of similar terms. Remember that you can only subtract and add similar terms, that is, terms with the same variable and the same indicator degrees. For example, you can add 7x and 5x, but you cannot add 7x and 5x 2 (since the exponents are different).

      • This rule also applies to members with multiple variables. For example, you can add 2xy 2 and -3xy 2 , but you cannot add 2xy 2 and -3x 2 y or 2xy 2 and -3y 2 .
      • Let's look at an example: x 2 + 3x + 6 - 8x. Here the like terms are 3x and 8x, so they can be added together. A simplified expression looks like this: x 2 - 5x + 6.

   2. Simplify the number fraction. In such a fraction, both the numerator and the denominator contain numbers (without a variable). Numerical fraction simplified in several ways. First, simply divide the denominator by the numerator. Second, factor the numerator and denominator and cancel the like factors (since dividing a number by itself will give you 1). In other words, if both the numerator and denominator have the same factor, you can drop it and get a simplified fraction.

      • For example, consider the fraction 36/60. Using a calculator, divide 36 by 60 to get 0.6. But you can simplify this fraction in another way by factoring the numerator and denominator: 36/60 = (6x6)/(6x10) = (6/6)*(6/10). Since 6/6 = 1, the simplified fraction is: 1 x 6/10 = 6/10. But this fraction can also be simplified: 6/10 = (2x3)/(2*5) = (2/2)*(3/5) = 3/5.

   3. If a fraction contains a variable, you can cancel like factors with the variable. Factor both the numerator and denominator and cancel the like factors, even if they contain the variable (remember that the like factors here may or may not contain the variable).

      • Let's look at an example: (3x 2 + 3x)/(-3x 2 + 15x). This expression can be rewritten (factored) in the form: (x + 1)(3x)/(3x)(5 - x). Since the 3x term is in both the numerator and denominator, you can cancel it out to give a simplified expression: (x + 1)/(5 - x). Let's look at another example: (2x 2 + 4x + 6)/2 = (2(x 2 + 2x + 3))/2 = x 2 + 2x + 3.
      • Please note that you cannot cancel any terms - only identical factors that are present in both the numerator and denominator are canceled. For example, in the expression (x(x + 2))/x, the variable (factor) “x” is in both the numerator and the denominator, so “x” can be reduced to obtain a simplified expression: (x + 2)/1 = x + 2. However, in the expression (x + 2)/x, the variable “x” cannot be reduced (since “x” is not a factor in the numerator).

   4. Open parenthesis. To do this, multiply the term outside the brackets by each term in the brackets. Sometimes this helps to simplify a complex expression. This applies to both members who are prime numbers, and to members that contain the variable.

      • For example, 3(x 2 + 8) = 3x 2 + 24, and 3x(x 2 + 8) = 3x 3 + 24x.
      • Please note that in fractional expressions There is no need to open the brackets if the same factor is present in both the numerator and the denominator. For example, in the expression (3(x 2 + 8))/3x there is no need to expand the parentheses, since here you can cancel the factor of 3 and get the simplified expression (x 2 + 8)/x. This expression is easier to work with; if you opened the parentheses, you would get the following complex expression: (3x 3 + 24x)/3x.

   5. Factor polynomials. Using this method, you can simplify some expressions and polynomials. Factoring is an operation opposite of opening brackets, that is, the expression is written as the product of two expressions, each of which is enclosed in brackets. In some cases, factorization can reduce same expression. IN special cases(usually with quadratic equations) factoring will allow you to solve the equation.

      • Consider the expression x 2 - 5x + 6. It is factored: (x - 3)(x - 2). Thus, if, for example, the expression is given (x 2 - 5x + 6)/(2(x - 2)), then you can rewrite it as (x - 3)(x - 2)/(2(x - 2)), reduce the expression (x - 2) and obtain a simplified expression (x - 3)/2.
      • Factoring polynomials is used to solve (find roots) equations (an equation is a polynomial equal to 0). For example, consider the equation x 2 - 5x + 6 = 0. By factoring it, you get (x - 3)(x - 2) = 0. Since any expression multiplied by 0 is equal to 0, we can write it like this : x - 3 = 0 and x - 2 = 0. Thus, x = 3 and x = 2, that is, you have found two roots of the equation given to you.

A literal expression (or an expression with variables) is mathematical expression, which consists of numbers, letters and symbols of mathematical operations. For example, the following expression is literal:

a+b+4

Using alphabetic expressions you can write laws, formulas, equations and functions. The ability to manipulate letter expressions is the key good knowledge algebra and higher mathematics.

Any serious problem in mathematics comes down to solving equations. And in order to be able to solve equations, you need to be able to work with literal expressions.

To work with literal expressions, you need to be well-versed in basic arithmetic: addition, subtraction, multiplication, division, basic laws of mathematics, fractions, operations with fractions, proportions. And not just study, but understand thoroughly.

Lesson content

Variables

Letters that are contained in literal expressions are called variables. For example, in the expression a+b+4 the variables are the letters a And b. If we substitute any numbers instead of these variables, then the literal expression a+b+4 contact numeric expression, the value of which can be found.

Numbers that are substituted for variables are called values ​​of variables. For example, let's change the values ​​of the variables a And b. The equal sign is used to change values

a = 2, b = 3

We have changed the values ​​of the variables a And b. Variable a assigned a value 2 , variable b assigned a value 3 . As a result, the literal expression a+b+4 turns into a regular numeric expression 2+3+4 whose value can be found:

2 + 3 + 4 = 9

When variables are multiplied, they are written together. For example, record ab means the same as the entry a×b. If we substitute the variables a And b numbers 2 And 3 , then we get 6

2 × 3 = 6

You can also write together the multiplication of a number by an expression in parentheses. For example, instead of a×(b + c) can be written down a(b + c). Applying the distribution law of multiplication, we obtain a(b + c)=ab+ac.

Odds

In literal expressions you can often find a notation in which a number and a variable are written together, for example 3a. This is actually a shorthand for multiplying the number 3 by a variable. a and this entry looks like 3×a .

In other words, the expression 3a is the product of the number 3 and the 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 A", or "increase the value of a variable a three times", but most often read as "three a«

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

3 × 5 = 15

Speaking in simple language, the coefficient is the number that comes before the letter (before the variable).

There can be several letters, for example 5abc. Here the coefficient is the number 5 . This coefficient shows that the product of variables abc increases fivefold. This expression can be read as " abc five times" or "increase the value of the expression abc five times" or "five abc«.

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

5 × 2 × 3 × 4 = 120

You can mentally imagine how the numbers 2, 3 and 4 were first multiplied, and the resulting value increased fivefold:

The sign of the coefficient refers only to the coefficient and does not apply to the variables.

Consider the expression −6b. Minus before the coefficient 6 , applies only to the coefficient 6 , and does not belong to the variable b. Understanding this fact will allow you not to make mistakes in the future with signs.

Let's find the value of the expression −6b at b = 3.

−6b −6×b. For clarity, let us write the expression −6b in expanded form and substitute the value of the variable b

−6b = −6 × b = −6 × 3 = −18

Example 2. Find the value of an expression −6b at b = −5

Let's write down the expression −6b in expanded form

−6b = −6 × b = −6 × (−5) = 30

Example 3. Find the value of an expression −5a+b at a = 3 And b = 2

−5a+b This short form entries from −5 × a + b, so for clarity we write the expression −5×a+b in expanded form and substitute the values ​​of the variables a And b

−5a + b = −5 × a + b = −5 × 3 + 2 = −15 + 2 = −13

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

but traditionally the unit is not written down, so they simply write a or ab

If there is a minus before the letter, then the coefficient is a number −1 . For example, the expression −a actually looks like −1a. This is the product of minus one and the variable a. It turned out like this:

−1 × a = −1a

There is a small catch here. In expression −a minus sign in front of the variable a actually refers to an "invisible unit" rather than a variable a. Therefore, you should be careful when solving problems.

For example, if given the expression −a and we are asked to find its value at a = 2, then at school we substituted a two instead of a variable a and received an answer −2 , without focusing too much on how it turned out. In fact, minus one was multiplied by positive number 2

−a = −1 × a

−1 × a = −1 × 2 = −2

If given the expression −a and you need to find its value at a = −2, then we substitute −2 instead of a variable a

−a = −1 × a

−1 × a = −1 × (−2) = 2

To avoid mistakes, at first invisible units can be written down explicitly.

Example 4. Find the value of an expression abc at a=2 , b=3 And c=4

Expression abc 1×a×b×c. For clarity, let us write the expression abc a, b And c

1 × a × b × c = 1 × 2 × 3 × 4 = 24

Example 5. Find the value of an expression abc at a=−2 , b=−3 And c=−4

Let's write down the expression abc in expanded form and substitute the values ​​of the variables a, b And c

1 × a × b × c = 1 × (−2) × (−3) × (−4) = −24

Example 6. Find the value of an expression − abc at a=3 , b=5 and c=7

Expression − abc this is a short form for −1×a×b×c. For clarity, let us write the expression − abc in expanded form and substitute the values ​​of the variables a, b And c

−abc = −1 × a × b × c = −1 × 3 × 5 × 7 = −105

Example 7. Find the value of an expression − abc at a=−2 , b=−4 and c=−3

Let's write down the expression − abc in expanded form:

−abc = −1 × a × b × c

Let's substitute the values ​​of the variables a , b And c

−abc = −1 × a × b × c = −1 × (−2) × (−4) × (−3) = 24

How to determine the coefficient

Sometimes you need to solve a problem in which you need to determine the coefficient of an expression. Basically, this task very simple. It is enough to be able to multiply numbers correctly.

To determine the coefficient in an expression, you need to separately multiply the numbers included in this expression and separately multiply the letters. The resulting numerical factor will be the coefficient.

Example 1. 7m×5a×(−3)×n

The expression consists of several factors. This can be clearly seen if you write the expression in expanded form. That is, the works 7m And 5a write it in the form 7×m And 5×a

7 × m × 5 × a × (−3) × n

Let's apply the associative law of multiplication, which allows you to multiply factors in any order. Namely, we will separately multiply the numbers and separately multiply the letters (variables):

−3 × 7 × 5 × m × a × n = −105man

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

−105amn

Example 2. Determine the coefficient in the expression: −a×(−3)×2

−a × (−3) × 2 = −3 × 2 × (−a) = −6 × (−a) = 6a

The coefficient is 6.

Example 3. Determine the coefficient in the expression:

Let's multiply numbers and letters separately:

The coefficient is −1. Please note that the unit is not written down, since it is customary not to write the coefficient 1.

These seemingly simplest tasks can play a very cruel joke on us. It often turns out that the sign of the coefficient is set incorrectly: either the minus is missing or, on the contrary, it was set in vain. To avoid these annoying mistakes, must be studied at a good level.

Addends in literal expressions

When adding several numbers, the sum of these numbers is obtained. Numbers that add are called addends. There can be several terms, for example:

1 + 2 + 3 + 4 + 5

When an expression consists of terms, it is much easier to evaluate because adding is easier than subtracting. But the expression can contain not only addition, but also subtraction, for example:

1 + 2 − 3 + 4 − 5

In this expression, the numbers 3 and 5 are subtrahends, not addends. But nothing prevents us from replacing subtraction with addition. Then we again get an expression consisting of terms:

1 + 2 + (−3) + 4 + (−5)

It doesn’t matter that the numbers −3 and −5 now have a minus sign. The main thing is that all the numbers in this expression are connected by an addition sign, that is, the expression is a sum.

Both expressions 1 + 2 − 3 + 4 − 5 And 1 + 2 + (−3) + 4 + (−5) equal to the same value - minus one

1 + 2 − 3 + 4 − 5 = −1

1 + 2 + (−3) + 4 + (−5) = −1

Thus, the meaning of the expression will not suffer if we replace subtraction with addition somewhere.

You can also replace subtraction with addition in literal expressions. For example, consider the following expression:

7a + 6b − 3c + 2d − 4s

7a + 6b + (−3c) + 2d + (−4s)

For any values ​​of 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 a teacher at school or a teacher at an institute may call even numbers (or variables) that are not addends.

For example, if the difference is written on the board a−b, then the teacher will not say that a is a minuend, and b- subtractable. He will call both variables one in general terms — terms. And all because the expression of the form a−b the mathematician sees how the sum a+(−b). In this case, the expression becomes a sum, and the variables a And (−b) become terms.

Similar terms

Similar terms- these are terms that have the same letter part. For example, consider the expression 7a + 6b + 2a. Components 7a And 2a have the same letter part - variable a. So the terms 7a And 2a are similar.

Typically, similar terms are added to simplify an expression or solve an equation. This operation is called bringing similar terms.

To bring similar terms, you need to add the coefficients of these terms, and multiply the resulting result by the common letter part.

For example, let us present similar terms in the expression 3a + 4a + 5a. In this case, all terms are similar. Let's add up their coefficients and multiply the result by the common letter part - by the variable a

3a + 4a + 5a = (3 + 4 + 5)×a = 12a

Similar terms are usually brought up in mind and the result is written down immediately:

3a + 4a + 5a = 12a

Also, one can reason as follows:

There were 3 variables a , 4 more variables a and 5 more variables a were added to them. As a result, we got 12 variables a

Let's look at several examples of bringing similar terms. Considering that this topic is very important, at first we will write down every little detail in detail. Even though everything is very simple here, most people make many mistakes. Mainly due to inattention, not ignorance.

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

Let's add up the coefficients in this expression and multiply the resulting result by the common letter part:

3a + 2a + 6a + 8a = (3 + 2 + 6 + 8) × a = 19a

design (3 + 2 + 6 + 8)×a You don’t have to write it down, so we’ll write down the answer right away

3a + 2a + 6a + 8a = 19a

Example 2. Give similar terms in the expression 2a+a

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

2a + 1a

Now let's present similar terms. That is, we add up the coefficients and multiply the result by the common letter part:

2a + 1a = (2 + 1) × a = 3a

Let's write down the solution briefly:

2a + a = 3a

2a+a, you can think differently:

Example 3. Give similar terms in the expression 2a−a

Let's replace subtraction with addition:

2a + (−a)

Second term (−a) written without a coefficient, but in reality it looks like (−1a). Coefficient −1 again invisible due to the fact that it is not recorded. So the expression looks like this:

2a + (−1a)

Now let's present similar terms. Let's add the coefficients and multiply the result by the common letter part:

2a + (−1a) = (2 + (−1)) × a = 1a = a

Usually written shorter:

2a − a = a

Giving similar terms in the expression 2a−a You can think differently:

There were 2 variables a, subtract one variable a, and as a result there was only one variable a left

Example 4. Give similar terms in the expression 6a − 3a + 4a − 8a

6a − 3a + 4a − 8a = 6a + (−3a) + 4a + (−8a)

Now let's present similar terms. Let's add the coefficients and multiply the result by the total letter part

(6 + (−3) + 4 + (−8)) × a = −1a = −a

Let's write down the solution briefly:

6a − 3a + 4a − 8a = −a

There are expressions that contain several various groups similar terms. For example, 3a + 3b + 7a + 2b. For such expressions, the same rules apply as for the others, namely, adding the coefficients and multiplying the result by the common letter part. But to avoid mistakes, it’s convenient different groups The terms are highlighted with different lines.

For example, in the expression 3a + 3b + 7a + 2b those terms that contain a variable a, can be underlined with one line, and those terms that contain a variable b, can be emphasized with two lines:

Now we can present similar terms. That is, add the coefficients and multiply the resulting result by the total letter part. This must be done for both groups of terms: for terms containing a variable a and for terms containing a variable b.

3a + 3b + 7a + 2b = (3+7)×a + (3 + 2)×b = 10a + 5b

Again, we repeat, the expression is simple, and similar terms can be given in mind:

3a + 3b + 7a + 2b = 10a + 5b

Example 5. Give similar terms in the expression 5a − 6a −7b + b

Let's replace subtraction with addition where possible:

5a − 6a −7b + b = 5a + (−6a) + (−7b) + b

Let us underline similar terms with different lines. Terms containing variables a we underline with one line, and the terms are the contents of the variables b, underline with two lines:

Now we can present similar terms. That is, add the coefficients and multiply the resulting result by the common letter part:

5a + (−6a) + (−7b) + b = (5 + (−6))×a + ((−7) + 1)×b = −a + (−6b)

If the expression contains regular numbers without letter factors, they are added separately.

Example 6. Give similar terms in the expression 4a + 3a − 5 + 2b + 7

Let's replace subtraction with addition where possible:

4a + 3a − 5 + 2b + 7 = 4a + 3a + (−5) + 2b + 7

Let us present similar terms. Numbers −5 And 7 do not have letter factors, but they are similar terms - they just need to be added. And the term 2b will remain unchanged, since it is the only one in this expression that has a letter factor b, and there is nothing to add it with:

4a + 3a + (−5) + 2b + 7 = (4 + 3)×a + 2b + (−5) + 7 = 7a + 2b + 2

Let's write down the solution briefly:

4a + 3a − 5 + 2b + 7 = 7a + 2b + 2

The terms can be ordered so that those terms that have the same letter part are located in the same part of the expression.

Example 7. Give similar terms in the expression 5t+2x+3x+5t+x

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

5t + 5t + 2x + 3x + x

Now we can present similar terms:

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

Let's write down the solution briefly:

5t + 2x + 3x + 5t + x = 10t + 6x

Sum opposite numbers equal to zero. This rule also works for literal expressions. If the expression contains identical terms, but with opposite signs, then you can get rid of them at the stage of reducing similar terms. In other words, simply eliminate them from the expression, since their sum is zero.

Example 8. Give similar terms in the expression 3t − 4t − 3t + 2t

Let's replace subtraction with addition where possible:

3t − 4t − 3t + 2t = 3t + (−4t) + (−3t) + 2t

Components 3t And (−3t) are opposite. The sum of opposite terms is zero. If we remove this zero from the expression, the value of the expression will not change, so we will remove it. And we will remove it by simply crossing out the terms 3t And (−3t)

As a result, we will be left with the expression (−4t) + 2t. In this expression, you can add similar terms and get the final answer:

(−4t) + 2t = ((−4) + 2)×t = −2t

Let's write down the solution briefly:

Simplifying Expressions

"simplify the expression" and below is the expression that needs to be simplified. Simplify an expression means making it simpler and shorter.

In fact, we've already been simplifying expressions when we've reduced fractions. After reduction, the fraction became shorter and easier to understand.

Consider the following example. Simplify the expression.

This task can literally be understood as follows: “Apply any valid actions to this expression, but make it simpler.” .

In this case, you can reduce the fraction, namely, divide the numerator and denominator of the fraction by 2:

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

As a result, the fraction was simplified to 0.5.

The first question you need to ask yourself when deciding similar tasks, it should be “What can be done?” . Because there are actions that you can do, and there are actions that you cannot do.

Another important point The thing to remember is that the value of the expression should not change after simplifying the expression. Let's return to the expression. This expression represents a division that can be performed. Having performed this division, we get the value of this expression, which is equal to 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 we also tried to simplify the expression by calculating it. As a result, we received a final answer of 0.5.

Thus, no matter how we simplify the expression, the value of the resulting expressions is still equal to 0.5. This means that the simplification was carried out correctly at every stage. This is exactly what we should strive for when simplifying expressions - the meaning of the expression should not suffer from our actions.

It is often necessary to simplify literal expressions. The same simplification rules apply to them as for numerical expressions. You can perform any valid actions, as long as the value of the expression does not change.

Let's look at a few examples.

Example 1. Simplify an expression 5.21s × t × 2.5

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

5.21s × t × 2.5 = 5.21 × 2.5 × s × t = 13.025 × st = 13.025st

So the expression 5.21s × t × 2.5 simplified to 13,025st.

Example 2. Simplify an expression −0.4 × (−6.3b) × 2

Second piece (−6.3b) can be translated into a form understandable to us, namely written in the form ( −6,3)×b , then multiply the numbers separately and multiply the letters separately:

− 0,4 × (−6.3b) × 2 = − 0,4 × (−6.3) × b × 2 = 5.04b

So the expression −0.4 × (−6.3b) × 2 simplified to 5.04b

Example 3. Simplify an expression

Let's write this expression in more detail to clearly see where the numbers are and where the letters are:

Now let’s multiply the numbers separately and multiply the letters separately:

So the expression simplified to −abc. This solution can be written briefly:

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

A fraction can be reduced by selecting a factor in the numerator and denominator and reducing these factors by their largest common divisor. In other words, use in which we do not describe in detail what the numerator and denominator were divided into.

For example, in the numerator the factor is 12 and in the denominator the factor 4 can be reduced by 4. We keep the four in our mind, and dividing 12 and 4 by this four, we write down the answers next to these numbers, having first crossed them out

Now you can multiply the resulting small factors. In this case, there are few of them and you can multiply them in your mind:

Over time, you may find that when solving a particular problem, expressions begin to “get fat,” so it is advisable to get used to fast calculations. What can be calculated in the mind must be calculated in the mind. What can be quickly reduced must be reduced quickly.

Example 4. Simplify an expression

So the expression simplified to

Example 5. Simplify an expression

Let's multiply the numbers separately and the letters separately:

So the expression simplified to mn.

Example 6. Simplify an expression

Let's write this expression in more detail to clearly see where the numbers are and where the letters are:

Now let’s multiply the numbers separately and the letters separately. For ease of calculation, the decimal fraction −6.4 and mixed number can be converted to ordinary fractions:

So the expression simplified to

The solution for this example can be written much shorter. It will look like this:

Example 7. Simplify an expression

Let's multiply numbers separately and letters separately. For ease of calculation, a mixed number and decimals 0.1 and 0.6 can be converted to ordinary fractions:

So the expression simplified to abcd. If you skip the details, then this decision can be written much shorter:

Notice how the fraction has been reduced. New factors that are obtained as a result of reduction of previous factors are also allowed to be reduced.

Now let's talk about what not to do. When simplifying expressions, it is strictly forbidden to multiply numbers and letters if the expression is a sum and not a product.

For example, if you want to simplify the expression 5a+4b, then you cannot write it like this:

This is the same as if we were asked to add two numbers and we multiplied them instead of adding them.

When substituting any variable values a And b expression 5a +4b turns into an ordinary numerical expression. Let's assume that the variables a And b have the following meanings:

a = 2, b = 3

Then the value of the expression will be equal to 22

5a + 4b = 5 × 2 + 4 × 3 = 10 + 12 = 22

First, multiplication is performed, and then the results are added. And if we tried to simplify this expression by multiplying numbers and letters, we would get the following:

5a + 4b = 5 × 4 × a × b = 20ab

20ab = 20 × 2 × 3 = 120

It turns out a completely different meaning of the expression. In the first case it worked 22 , in the second case 120 . This means that simplifying the expression 5a+4b was performed incorrectly.

After simplifying the expression, its value should not change with the same values ​​of the variables. If, when substituting any variable values ​​into the original expression, one value is obtained, then after simplifying the expression, the same value should be obtained as before the simplification.

With expression 5a+4b there's really nothing you can do. It doesn't simplify it.

If an expression contains similar terms, then they can be added if our goal is to simplify the expression.

Example 8. Simplify an expression 0.3a−0.4a+a

0.3a − 0.4a + a = 0.3a + (−0.4a) + a = (0.3 + (−0.4) + 1)×a = 0.9a

or shorter: 0.3a − 0.4a + a = 0.9a

So the expression 0.3a−0.4a+a simplified to 0.9a

Example 9. Simplify an expression −7.5a − 2.5b + 4a

To simplify this expression, we can add similar terms:

−7.5a − 2.5b + 4a = −7.5a + (−2.5b) + 4a = ((−7.5) + 4)×a + (−2.5b) = −3.5a + (−2.5b)

or shorter −7.5a − 2.5b + 4a = −3.5a + (−2.5b)

Term (−2.5b) remained unchanged because there was nothing to put it with.

Example 10. Simplify an expression

To simplify this expression, we can add similar terms:

The coefficient was for ease of calculation.

So the expression simplified to

Example 11. Simplify an expression

To simplify this expression, we can add similar terms:

So the expression simplified to .

IN in this example It would be more appropriate to add the first and last coefficients first. In this case we would have a short solution. It would look like this:

Example 12. Simplify an expression

To simplify this expression, we can add similar terms:

So the expression simplified to .

The term remained unchanged, since there was nothing to add it to.

This solution can be written much shorter. It will look like this:

The short solution skipped the steps of replacing subtraction with addition and detailing how fractions were reduced to a common denominator.

Another difference is that in detailed solution the answer looks like , but in short as . In fact, they are the same expression. The difference is that in the first case, subtraction is replaced by addition, since at the beginning when we wrote the solution in in detail, we replaced subtraction with addition wherever possible, and this replacement was preserved for the answer.

Identities. Identically equal expressions

Once we have simplified any expression, it becomes simpler and shorter. To check whether the simplified expression is correct, it is enough to substitute any variable values ​​first into the previous expression that needed to be simplified, and then into the new one that was simplified. If the value in both expressions is the same, then the simplified expression is true.

Let's consider simplest example. Let it be necessary to simplify the expression 2a×7b. To simplify this expression, you can multiply numbers and letters separately:

2a × 7b = 2 × 7 × a × b = 14ab

Let's check whether we simplified the expression correctly. To do this, let’s substitute any values ​​of the variables a And b first into the first expression that needed to be simplified, and then into the second, which was simplified.

Let the values ​​of the variables a , b will be as follows:

a = 4, b = 5

Let's substitute them into the first expression 2a×7b

Now let’s substitute the same variable values ​​into the expression that resulted from simplification 2a×7b, namely in the expression 14ab

14ab = 14 × 4 × 5 = 280

We see that when a=4 And b=5 value of the first expression 2a×7b and the meaning of the second expression 14ab equal

2a × 7b = 2 × 4 × 7 × 5 = 280

14ab = 14 × 4 × 5 = 280

The same will happen for any other values. For example, let a=1 And b=2

2a × 7b = 2 × 1 × 7 × 2 =28

14ab = 14 × 1 × 2 =28

Thus, for any values expression variables 2a×7b And 14ab are equal to the same value. Such expressions are called identically equal.

We conclude that between the expressions 2a×7b And 14ab you can put an equal sign because they are equal to the same value.

2a × 7b = 14ab

An equality is any expression that is connected by an equal sign (=).

And equality of the form 2a×7b = 14ab called identity.

An identity is an equality that is true for any values ​​of the variables.

Other examples of identities:

a + b = b + a

a(b+c) = ab + ac

a(bc) = (ab)c

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

Faithful numerical equalities are also identities. For example:

2 + 2 = 4

3 + 3 = 5 + 1

10 = 7 + 2 + 1

Deciding difficult task To make the calculation easier, the complex expression is replaced with a simpler expression that is identically equal to the previous one. This replacement is called identical transformation of the expression or simply transforming the expression.

For example, we simplified the expression 2a×7b, and got a simpler expression 14ab. This simplification can be called the identity transformation.

You can often find a task that says "prove that equality is an identity" and then the equality that needs to be proven is given. Usually this equality consists of two parts: the left and right parts of the equality. Our task is to perform identity transformations with one of the parts of the equality and obtain the other part. Or perform identical transformations on both sides of the equality and make sure that both sides of the equality contain the same expressions.

For example, let us prove that the equality 0.5a × 5b = 2.5ab is an identity.

Let's simplify the left side of this equality. To do this, multiply the numbers and letters separately:

0.5 × 5 × a × b = 2.5ab

2.5ab = 2.5ab

As a result of a small identity transformation, left side equality became equal to the right side of the equality. So we have proven that the equality 0.5a × 5b = 2.5ab is an identity.

From identical transformations we learned to add, subtract, multiply and divide numbers, reduce fractions, add similar terms, and also simplify some expressions.

But these are not all identical transformations that exist in mathematics. Identity transformations a lot more. We will see this more than once in the future.

Tasks for independent solution:

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Section 5 EXPRESSIONS AND EQUATIONS

In this section you will learn:

ü o expressions and their simplifications;

ü what are the properties of equalities;

ü how to solve equations based on the properties of equalities;

ü what types of problems are solved using equations; what are perpendicular lines and how to build them;

ü what lines are called parallel and how to build them;

ü what is a coordinate plane?

ü how to determine the coordinates of a point on a plane;

ü what is a graph of the relationship between quantities and how to construct it;

ü how to apply the studied material in practice

§ 30. EXPRESSIONS AND THEIR SIMPLIFICATION

You already know what letter expressions are and know how to simplify them using the laws of addition and multiplication. For example, 2a ∙ (-4 b ) = -8 ab . In the resulting expression, the number -8 is called the coefficient of the expression.

Does the expression CD coefficient? So. It is equal to 1 because cd - 1 ∙ cd .

Recall that converting an expression with parentheses into an expression without parentheses is called expanding the parentheses. For example: 5(2x + 4) = 10x+ 20.

The reverse action in this example is to take the common factor out of brackets.

Terms containing the same letter factors are called similar terms. By taking the common factor out of brackets, similar terms are raised:

5x + y + 4 - 2x + 6 y - 9 =

= (5x - 2x) + (y + 6 y )+ (4 - 9) = = (5-2)* + (1 + 6)* y -5 =

B x+ 7y - 5.

Rules for opening parentheses

1. If there is a “+” sign in front of the brackets, then when opening the brackets, the signs of the terms in the brackets are preserved;

2. If there is a “-” sign in front of the brackets, then when the brackets are opened, the signs of the terms in the brackets change to the opposite.

Task 1. Simplify the expression:

1) 4x+(-7x + 5);

2) 15 y -(-8 + 7 y ).

Solutions. 1. Before the brackets there is a “+” sign, so when opening the brackets, the signs of all terms are preserved:

4x +(-7x + 5) = 4x - 7x + 5=-3x + 5.

2. Before the brackets there is a “-” sign, so when opening the brackets: the signs of all terms are reversed:

15 - (- 8 + 7y) = 15y + 8 - 7y = 8y +8.

To open parentheses use distributive property multiplication: a( b + c ) = ab + ac. If a > 0, then the signs of the terms b and with do not change. If a< 0, то знаки слагаемых b and change to the opposite.

Task 2. Simplify the expression:

1) 2(6 y -8) + 7 y ;

2)-5(2-5x) + 12.

Solutions. 1. The factor 2 in front of the brackets is positive, therefore, when opening the brackets, we preserve the signs of all terms: 2(6 y - 8) + 7 y = 12 y - 16 + 7 y =19 y -16.

2. The factor -5 in front of the brackets is negative, so when opening the brackets, we change the signs of all terms to the opposite:

5(2 - 5x) + 12 = -10 + 25x +12 = 2 + 25x.

Find out more

1. The word “sum” comes from Latin summa , which means “total”, “total amount”.

2. The word “plus” comes from Latin plus which means "more" and the word "minus" is from Latin minus What does "less" mean? The signs “+” and “-” are used to indicate the operations of addition and subtraction. These signs were introduced by the Czech scientist J. Widman in 1489 in the book “A quick and pleasant account for all merchants”(Fig. 138).

Rice. 138

REMEMBER THE IMPORTANT

1. What terms are called similar? How are such terms constructed?

2. How do you open parentheses preceded by a “+” sign?

3. How do you open parentheses preceded by a “-” sign?

4. How do you open parentheses preceded by a positive factor?

5. How do you open parentheses that are preceded by a negative factor?

1374". Name the coefficient of the expression:

1)12 a; 3) -5.6 xy;

2)4 6; 4)-s.

1375". Name the terms that differ only by coefficient:

1) 10a + 76-26 + a; 3) 5 n + 5 m -4 n + 4;

2) bc -4 d - bc + 4 d ; 4)5x + 4y-x + y.

What are these terms called?

1376". Is there similar terms in the expression:

1)11a+10a; 3)6 n + 15 n ; 5) 25r - 10r + 15r;

2) 14s-12; 4)12 m + m ; 6)8 k +10 k - n ?

1377". Is it necessary to change the signs of the terms in brackets, opening the brackets in the expression:

1)4 + (a+ 3 b); 2)-c +(5-d); 3) 16-(5 m -8 n)?

1378°. Simplify the expression and underline the coefficient:

1379°. Simplify the expression and underline the coefficient:

1380°. Combine similar terms:

1) 4a - Po + 6a - 2a; 4) 10 - 4 d - 12 + 4 d ;

2) 4 b - 5 b + 4 + 5 b ; 5) 5a - 12 b - 7a + 5 b;

3)-7 ang="EN-US">c+ 5-3 c + 2; 6) 14 n - 12 m -4 n -3 m.

1381°. Combine similar terms:

1) 6a - 5a + 8a -7a; 3) 5s + 4-2s-3s;

2)9 b +12-8-46; 4) -7 n + 8 m - 13 n - 3 m.

1382°. Take the common factor out of brackets:

1)1.2 a +1.2 b; 3) -3 n - 1.8 m; 5)-5 p + 2.5 k -0.5 t ;

2) 0.5 s + 5 d; 4) 1.2 n - 1.8 m; 6) -8r - 10k - 6t.

1383°. Take the common factor out of brackets:

1) 6a-12 b; 3) -1.8 n -3.6 m;

2) -0.2 s + 1 4 d ; A) 3p - 0.9 k + 2.7 t.

1384°. Open the brackets and combine similar terms;

1) 5 + (4a -4); 4) -(5 c - d) + (4 d + 5c);

2) 17x-(4x-5); 5) (n - m) - (-2 m - 3 n);

3) (76 - 4) - (46 + 2); 6) 7(-5x + y) - (-2y + 4x) + (x - 3y).

1385°. Open the brackets and combine similar terms:

1) 10a + (4 - 4a); 3) (s - 5 d) - (- d + 5c);

2) -(46- 10) + (4- 56); 4)-(5 n + m) + (-4 n + 8 m)-(2 m -5 n).

1386°. Open the brackets and find the meaning of the expression:

1)15+(-12+ 4,5); 3) (14,2-5)-(12,2-5);

2) 23-(5,3-4,7); 4) (-2,8 + 13)-(-5,6 + 2,8) + (2,8-13).

1387°. Open the brackets and find the meaning of the expression:

1) (14- 15,8)- (5,8 + 4);

2)-(18+22,2)+ (-12+ 22,2)-(5- 12).

1388°. Open parenthesis:

1)0.5 ∙ (a + 4); 4) (n - m) ∙ (-2.4 p);

2)-s ∙ (2.7-1.2 d ); 5)3 ∙ (-1.5 r + k - 0.2 t);

3) 1.6 ∙ (2 n + m); 6) (4.2 p - 3.5 k -6 t) ∙ (-2a).

1389°. Open parenthesis:

1) 2.2 ∙ (x-4); 3)(4 c - d )∙(-0.5 y );

2) -2 ∙ (1.2 n - m); 4)6- (-р + 0.3 k - 1.2 t).

1390. Simplify the expression:

1391. Simplify the expression:

1392. Reduce similar terms:

1393. Combine similar terms:

1394. Simplify the expression:

1)2.8 - (0.5 a + 4) - 2.5 ∙ (2a - 6);

2) -12 ∙ (8 - 2, by ) + 4.5 ∙ (-6 y - 3.2);

4) (-12.8 m + 24.8 n) ∙ (-0.5)-(3.5 m -4.05 m) ∙ 2.

1395. Simplify the expression:

1396. Find the meaning of the expression;

1) 4-(0.2 a-3)-(5.8 a-16), if a = -5;

2) 2-(7-56)+ 156-3∙(26+ 5), if = -0.8;

m = 0.25, n = 5.7.

1397. Find the meaning of the expression:

1) -4∙ (i-2) + 2∙(6x - 1), if x = -0.25;

1398*. Find the error in the solution:

1)5- (a-2.4)-7 ∙ (-a+ 1.2) = 5a - 12-7a + 8.4 = -2a-3.6;

2) -4 ∙ (2.3 a - 6) + 4.2 ∙ (-6 - 3.5 a) = -9.2 a + 46 + 4.26 - 14.7 a = -5.5 a + 8.26.

1399*. Open the parentheses and simplify the expression:

1) 2ab - 3(6(4a - 1) - 6(6 - 10a)) + 76;

1400*. Arrange the parentheses to get the correct equality:

1)a-6-a + 6 = 2a; 2) a -2 b -2 a + b = 3 a -3 b .

1401*. Prove that for any numbers a and b if a > b , then the equality holds:

1) (a + b) + (a- b) = 2a; 2) (a + b) - (a - b) = 2 b.

Will this equality be correct if: a) a< b ; b) a = 6?

1402*. Prove that for any natural number and the arithmetic mean of the previous and following numbers is equal to the number a.

PUT IT IN PRACTICE

1403. To prepare a fruit dessert for three people you need: 2 apples, 1 orange, 2 bananas and 1 kiwi. How to create a letter expression to determine the amount of fruit needed to prepare dessert for guests? Help Marin calculate how many fruits she needs to buy if: 1) 5 friends come to visit her; 2) 8 friends.

1404. Make a letter expression to determine the time required to complete your math homework if:

1) a min was spent on solving problems; 2) simplification of expressions is 2 times greater than for solving problems. How long did it take to complete homework Vasilko, if he spent 15 minutes solving problems?

1405. Lunch in the school cafeteria consists of salad, borscht, cabbage rolls and compote. The cost of salad is 20%, borscht - 30%, cabbage rolls - 45%, compote - 5% of the total cost of the entire lunch. Write an expression to find the cost of lunch in the school canteen. How much does lunch cost if the price of salad is 2 UAH?

REVIEW PROBLEMS

1406. Solve the equation:

1407. Tanya spent on ice creamall available money, and for candy -the rest. How much money does Tanya have left?

if candy costs 12 UAH?

Note 1

A Boolean function can be written using a Boolean expression and can then be moved to a logic circuit. It is necessary to simplify logical expressions in order to obtain the simplest (and therefore cheaper) logical circuit possible. Essentially, a logical function, a logical expression and logic circuit-that's three different languages, telling about one entity.

To simplify logical expressions use laws of algebra logic.

Some transformations are similar to transformations of formulas in classical algebra (taking the common factor out of brackets, using commutative and combinational laws etc.), and other transformations are based on properties that the operations of classical algebra do not possess (use of the distributive law for conjunction, laws of absorption, gluing, de Morgan's rules, etc.).

The laws of algebra of logic are formulated for basic logical operations- “NOT” – inversion (negation), “AND” – conjunction (logical multiplication) and “OR” – disjunction (logical addition).

The law of double negation means that the “NOT” operation is reversible: if you apply it twice, then in the end the logical value will not change.

The law of excluded middle states that any logical expression is either true or false (“there is no third”). Therefore, if $A=1$, then $\bar(A)=0$ (and vice versa), which means that the conjunction of these quantities is always equal to zero, and the disjunction is always equal to one.

$((A + B) → C) \cdot (B → C \cdot D) \cdot C.$

Let's simplify this formula:

Figure 3.

It follows that $A = 0$, $B = 1$, $C = 1$, $D = 1$.

Answer: Students $B$, $C$ and $D$ play chess, but student $A$ does not play.

When simplifying logical expressions, you can perform the following sequence of actions:

1. Replace all “non-basic” operations (equivalence, implication, exclusive OR, etc.) with their expressions through basic operations inversion, conjunction and disjunction.
2. Expand inversions complex expressions according to De Morgan's rules in such a way that negation operations remain only for individual variables.
3. Then simplify the expression using opening parentheses, removing common factors beyond brackets and other laws of algebra of logic.

Example 2

Here, De Morgan's rule, the distributive law, the law of the excluded middle, the commutative law, the law of repetition, again the commutative law and the law of absorption are used successively.