Using any language, you can express the same information in different words and phrases. Mathematical language is no exception. But the same expression can be equivalently written in different ways. And in some situations, one of the entries is simpler. We'll talk about simplifying expressions in this lesson.

People communicate on different languages. For us, an important comparison is the pair “Russian language - mathematical language”. The same information can be communicated in different languages. But, besides this, it can be pronounced in different ways in one language.

For example: “Petya is friends with Vasya”, “Vasya is friends with Petya”, “Petya and Vasya are friends”. Said differently, but the same thing. From any of these phrases we would understand what we are talking about.

Let's look at this phrase: “The boy Petya and the boy Vasya are friends.” We understand what we mean we're talking about. However, we don't like the sound of this phrase. Can't we simplify it, say the same thing, but simpler? “Boy and boy” - you can say once: “Boys Petya and Vasya are friends.”

“Boys”... Isn’t it clear from their names that they are not girls? We remove the “boys”: “Petya and Vasya are friends.” And the word “friends” can be replaced with “friends”: “Petya and Vasya are friends.” As a result, the first, long, ugly phrase was replaced with an equivalent statement that is easier to say and easier to understand. We have simplified this phrase. To simplify means to say it more simply, but not to lose or distort the meaning.

In mathematical language, roughly the same thing happens. One and the same thing can be said, written differently. What does it mean to simplify an expression? This means that for the original expression there are many equivalent expressions, that is, those that mean the same thing. And from all this variety we must choose the simplest, in our opinion, or the most suitable for our further purposes.

For example, consider the numeric expression . It will be equivalent to .

It will also be equivalent to the first two: .

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

For numeric expressions, you always need to do everything and get the equivalent expression as a single number.

Let's look at an example of a literal expression . Obviously, it will be simpler.

When simplifying literal expressions, it is necessary to perform all possible actions.

Is it always necessary to simplify an expression? No, sometimes it will be more convenient for us to have an equivalent but longer entry.

Example: you need to subtract a number from a number.

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

That is, a simplified expression is not always beneficial for us for further calculations.

Nevertheless, very often we are faced with a task that just sounds like “simplify the expression.”

Simplify the expression: .

Solution

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

2) Let's calculate the products: .

Obviously, the last expression has a simpler form than the initial one. We've simplified it.

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

To determine the equivalent expression you need:

1) perform all possible actions,

2) use the properties of addition, subtraction, multiplication and division to simplify calculations.

Properties of addition and subtraction:

1. Commutative property of addition: rearranging the terms does not change the sum.

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

3. The property of subtracting a sum from a number: to subtract a sum from a number, you can subtract each term separately.

Properties of multiplication and division

1. Commutative property of multiplication: rearranging the factors does not change the product.

2. Combinative property: to multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second factor.

3. Distributive property of multiplication: in order to multiply a number by a sum, you need to multiply it by each term separately.

Let's see how we actually do mental calculations.

Calculate:

Solution

1) Let's imagine how

2) Let's imagine the first factor as a sum of bit terms and perform the multiplication:

3) you can imagine how and perform multiplication:

4) Replace the first factor with an equivalent sum:

The distributive law can also be used in reverse side: .

Follow these steps:

1) 2)

Solution

1) For convenience, you can use the distributive law, only use it in the opposite direction - take the common factor out of brackets.

2) Let’s take the common factor out of brackets

It is necessary to buy linoleum for the kitchen and hallway. Kitchen area - , hallway - . There are three types of linoleums: for, and rubles for. How much will each of the three types of linoleum cost? (Fig. 1)

Rice. 1. Illustration for the problem statement

Solution

Method 1. You can separately find out how much money it will take to buy linoleum for the kitchen, and then put it in the hallway and add up the resulting products.

First level

Converting Expressions. Detailed theory (2019)

Converting Expressions

We often hear this unpleasant phrase: “simplify the expression.” Usually we see some kind of monster like this:

“It’s much simpler,” we say, but such an answer usually doesn’t work.

Now I will teach you not to be afraid of any such tasks. Moreover, at the end of the lesson, you yourself will simplify this example to (just!) an ordinary number (yes, to hell with these letters).

But before you start this lesson, you need to be able to handle fractions and factor polynomials. Therefore, first, if you have not done this before, be sure to master the topics “” and “”.

Have you read it? If yes, then you are now ready.

Basic simplification operations

Now let's look at the basic techniques that are used to simplify expressions.

The simplest one is

1. Bringing similar

What are similar? You took this in 7th grade, when letters instead of numbers first appeared in mathematics. Similar are terms (monomials) with the same letter part. For example, in the sum, similar terms are and.

Do you remember?

To bring similar means to add several similar terms to each other and get one term.

How can we put the letters together? - you ask.

This is very easy to understand if you imagine that the letters are some kind of objects. For example, a letter is a chair. Then what is the expression equal to? Two chairs plus three chairs, how many will it be? That's right, chairs: .

Now try this expression: .

To avoid confusion, let different letters represent different objects. For example, - is (as usual) a chair, and - is a table. Then:

chairs tables chair tables chairs chairs tables

The numbers by which the letters in such terms are multiplied are called coefficients. For example, in a monomial the coefficient is equal. And in it is equal.

So, the rule for bringing similar ones is:

Examples:

Give similar ones:

Answers:

2. (and similar, since, therefore, these terms have the same letter part).

2. Factorization

This is usually the most an important part in simplifying expressions. After you have given similar ones, most often the resulting expression needs to be factorized, that is, presented as a product. This is especially important in fractions: in order to be able to reduce a fraction, the numerator and denominator must be represented as a product.

You went through the methods of factoring expressions in detail in the topic “”, so here you just have to remember what you learned. To do this, decide a few examples(needs to be factorized):

Solutions:

3. Reducing a fraction.

Well, what could be more pleasant than crossing out part of the numerator and denominator and throwing them out of your life?

That's the beauty of downsizing.

It's simple:

If the numerator and denominator contain the same factors, they can be reduced, that is, removed from the fraction.

This rule follows from the basic property of a fraction:

That is, the essence of the reduction operation is that We divide the numerator and denominator of the fraction by the same number (or by the same expression).

To reduce a fraction you need:

1) numerator and denominator factorize

2) if the numerator and denominator contain common factors, they can be crossed out.

The principle, I think, is clear?

I would like to draw your attention to one thing typical mistake when contracting. Although this topic is simple, many people do everything wrong, not understanding that reduce- this means divide numerator and denominator are the same number.

No abbreviations if the numerator or denominator is a sum.

For example: we need to simplify.

Some people do this: which is absolutely wrong.

Another example: reduce.

The “smartest” will do this: .

Tell me what's wrong here? It would seem: - this is a multiplier, which means it can be reduced.

But no: - this is a factor of only one term in the numerator, but the numerator itself as a whole is not factorized.

Here's another example: .

This expression is factorized, which means you can reduce it, that is, divide the numerator and denominator by, and then by:

You can immediately divide it into:

To avoid such mistakes, remember easy way how to determine whether an expression is factorized:

The arithmetic operation that is performed last when calculating the value of an expression is the “master” operation. That is, if you substitute some (any) numbers instead of letters and try to calculate the value of the expression, then if the last action is multiplication, then we have a product (the expression is factorized). If the last action is addition or subtraction, this means that the expression is not factorized (and therefore cannot be reduced).

To consolidate, solve a few yourself examples:

Answers:

1. I hope you didn’t immediately rush to cut and? It was still not enough to “reduce” units like this:

The first step should be factorization:

4. Adding and subtracting fractions. Reducing fractions to a common denominator.

Adding and subtracting ordinary fractions is a familiar operation: we look for a common denominator, multiply each fraction by the missing factor and add/subtract the numerators. Let's remember:

Answers:

1. The denominators and are relatively prime, that is, they do not have common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:

2. Here the common denominator is:

3. First thing here mixed fractions we turn them into incorrect ones, and then follow the usual pattern:

It's a completely different matter if the fractions contain letters, for example:

Let's start with something simple:

a) Denominators do not contain letters

Here everything is the same as with ordinary numerical fractions: we find the common denominator, multiply each fraction by the missing factor and add/subtract the numerators:

Now in the numerator you can give similar ones, if any, and factor them:

Try it yourself:

b) Denominators contain letters

Let's remember the principle of finding a common denominator without letters:

· first of all, we determine the common factors;

· then we write out all the common factors one at a time;

· and multiply them by all other non-common factors.

To determine the common factors of the denominators, we first factor them into prime factors:

Let us emphasize the common factors:

Now let’s write out the common factors one at a time and add to them all the non-common (not underlined) factors:

This is the common denominator.

Let's get back to the letters. The denominators are given in exactly the same way:

· factor the denominators;

· determine common (identical) factors;

· write out all common factors once;

· multiply them by all other non-common factors.

So, in order:

1) factor the denominators:

2) determine common (identical) factors:

3) write out all the common factors once and multiply them by all other (non-underlined) factors:

So there's a common denominator here. The first fraction must be multiplied by, the second - by:

By the way, there is one trick:

For example: .

We see the same factors in the denominators, only all with different indicators. The common denominator will be:

to a degree

to a degree

to a degree

to a degree.

Let's complicate the task:

How to make fractions have the same denominator?

Let's remember the basic property of a fraction:

Nowhere does it say that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because it's not true!

See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example, . What did you learn?

So, another unshakable rule:

When you reduce fractions to a common denominator, use only the multiplication operation!

But what do you need to multiply by to get?

So multiply by. And multiply by:

We will call expressions that cannot be factorized “elementary factors.” For example, - this is an elementary factor. - Same. But no: it can be factorized.

What about the expression? Is it elementary?

No, because it can be factorized:

(you already read about factorization in the topic “”).

So, the elementary factors into which you decompose an expression with letters are an analogue of the simple factors into which you decompose numbers. And we will deal with them in the same way.

We see that both denominators have a multiplier. It will go to the common denominator to the degree (remember why?).

The factor is elementary, and they do not have a common factor, which means that the first fraction will simply have to be multiplied by it:

Another example:

Solution:

Before you multiply these denominators in a panic, you need to think about how to factor them? They both represent:

Great! Then:

Another example:

Solution:

As usual, let's factorize the denominators. In the first denominator we simply put it out of brackets; in the second - the difference of squares:

It would seem that there are no common factors. But if you look closely, they are similar... And it’s true:

So let's write:

That is, it turned out like this: inside the bracket we swapped the terms, and at the same time the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.

Now let's bring it to a common denominator:

Got it? Let's check it now.

Tasks for independent solution:

Answers:

Here we need to remember one more thing - the difference of cubes:

Please note that the denominator of the second fraction does not contain the formula “square of the sum”! The square of the sum would look like this: .

A is the so-called incomplete square of the sum: the second term in it is the product of the first and last, and not their double product. Incomplete square the sum is one of the factors in the expansion of the difference of cubes:

What to do if there are already three fractions?

Yes, the same thing! First of all, let’s make sure that the maximum number of factors in the denominators is the same:

Please note: if you change the signs inside one bracket, the sign in front of the fraction changes to the opposite. When we change the signs in the second bracket, the sign in front of the fraction changes again to the opposite. As a result, it (the sign in front of the fraction) has not changed.

We write out the entire first denominator into the common denominator, and then add to it all the factors that have not yet been written, from the second, and then from the third (and so on, if there are more fractions). That is, it turns out like this:

Hmm... It’s clear what to do with fractions. But what about the two?

It's simple: you know how to add fractions, right? So, we need to make two become a fraction! Let's remember: a fraction is a division operation (the numerator is divided by the denominator, in case you forgot). And there is nothing easier than dividing a number by. In this case, the number itself will not change, but will turn into a fraction:

Exactly what is needed!

5. Multiplication and division of fractions.

Well, the hardest part is over now. And ahead of us is the simplest, but at the same time the most important:

Procedure

What is the procedure for calculating a numerical expression? Remember by calculating the meaning of this expression:

Did you count?

It should work.

So, let me remind you.

The first step is to calculate the degree.

The second is multiplication and division. If there are several multiplications and divisions at the same time, they can be done in any order.

And finally, we perform addition and subtraction. Again, in any order.

But: the expression in brackets is evaluated out of turn!

If several brackets are multiplied or divided by each other, we first calculate the expression in each of the brackets, and then multiply or divide them.

What if there are more brackets inside the brackets? Well, let's think: some expression is written inside the brackets. When calculating an expression, what should you do first? That's right, calculate the brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.

So, the procedure for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):

Okay, it's all simple.

But this is not the same as an expression with letters?

No, it's the same! Only instead of arithmetic operations you need to do algebraic ones, that is, the actions described in previous section: bringing similar, adding fractions, reducing fractions, and so on. The only difference will be the action of factoring polynomials (we often use this when working with fractions). Most often, to factorize, you need to use I or simply put the common factor out of brackets.

Usually our goal is to represent the expression as a product or quotient.

For example:

Let's simplify the expression.

1) First, we simplify the expression in brackets. There we have a difference of fractions, and our goal is to present it as a product or quotient. So, we bring the fractions to a common denominator and add:

It is impossible to simplify this expression any further; all the factors here are elementary (do you still remember what this means?).

2) We get:

Multiplying fractions: what could be simpler.

3) Now you can shorten:

OK it's all over Now. Nothing complicated, right?

Another example:

Simplify the expression.

First, try to solve it yourself, and only then look at the solution.

First of all, let's determine the order of actions. First, let's add the fractions in parentheses, so instead of two fractions we get one. Then we will do division of fractions. Well, let's add the result with the last fraction. I will number the steps schematically:

Now I’ll show you the process, tinting the current action in red:

Finally, I will give you two useful tips:

1. If there are similar ones, they must be brought immediately. At whatever point similar ones arise in our country, it is advisable to bring them up immediately.

2. The same applies to reducing fractions: as soon as the opportunity to reduce appears, it must be taken advantage of. The exception is for fractions that you add or subtract: if they now have the same denominators, then the reduction should be left for later.

Here are some tasks for you to solve on your own:

And what was promised at the very beginning:

Solutions (brief):

If you have coped with at least the first three examples, then you have mastered the topic.

Now on to learning!

CONVERTING EXPRESSIONS. SUMMARY AND BASIC FORMULAS

Basic simplification operations:

• Bringing similar: to add (reduce) similar terms, you need to add their coefficients and assign the letter part.
• Factorization: putting the common factor out of brackets, applying it, etc.
• Reducing a fraction: The numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, which does not change the value of the fraction.
  1) numerator and denominator factorize
  2) if the numerator and denominator have common factors, they can be crossed out.

  IMPORTANT: only multipliers can be reduced!

• Adding and subtracting fractions:
  ;
• Multiplying and dividing fractions:
  ;

§ 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 letter expression is to make it shorter, with a minimum number of 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:

A literal expression (or variable expression) is a mathematical expression that consists of numbers, letters, and mathematical symbols. 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 to good knowledge of 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 will turn into a numerical expression whose value 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 is a short form for −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 the 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. In principle, this task is 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, it 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 to mind and the result is written down immediately:

3a + 4a + 5a = 12a

Also, one can reason as follows:

There were 3 a variables, 4 more a variables and 5 more a variables 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 different groups of 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 in order to avoid mistakes, it is convenient to highlight different groups of terms 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 ordinary numbers without letter factors, then 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

The sum of opposite numbers is 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 solving such problems 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 the numbers separately and multiply the 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 both the numerator and the denominator and reducing these factors by their greatest common factor. 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 quick 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 a mixed number can be converted into 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, this solution 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 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, because at the beginning, when we wrote down the solution in detailed form, 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 ​​of the 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.

True 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, the left side of the 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. There are many more identical transformations. We will see this more than once in the future.

Tasks for independent solution:

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An algebraic expression in which, along with the operations of addition, subtraction and multiplication, also uses division into letter expressions, is called a fractional algebraic expression. These are, for example, the expressions

We call an algebraic fraction an algebraic expression that has the form of a quotient of the division of two integer algebraic expressions (for example, monomials or polynomials). These are, for example, the expressions

The third of the expressions).

Identical transformations of fractional algebraic expressions are for the most part intended to represent them in the form algebraic fraction. To find the common denominator, factorization of the denominators of fractions is used - terms in order to find their least common multiple. When reducing algebraic fractions, the strict identity of expressions may be violated: it is necessary to exclude values ​​of quantities at which the factor by which the reduction is made becomes zero.

Let us give examples of identical transformations of fractional algebraic expressions.

Example 1: Simplify an expression

All terms can be reduced to a common denominator (it is convenient to change the sign in the denominator of the last term and the sign in front of it):

Our expression is equal to one for all values ​​except these values; it is undefined and reducing the fraction is illegal).

Example 2. Represent the expression as an algebraic fraction

Solution. The expression can be taken as a common denominator. We find sequentially:

Exercises

1. Find the values ​​of algebraic expressions for the specified parameter values:

2. Factorize.