Rules for finding the value of a numerical expression. Numeric and algebraic expressions

As a rule, children begin to study algebra in elementary school. After mastering the basic principles of working with numbers, they solve examples with one or more unknown variables. Finding the meaning of an expression like this can be quite difficult, but if you simplify it using elementary school knowledge, everything will work out quickly and easily.

What is the meaning of an expression

A numerical expression is an algebraic notation consisting of numbers, parentheses and signs if it makes sense.

In other words, if it is possible to find the meaning of an expression, then the entry is not without meaning, and vice versa.

Examples following entries are correct numeric constructions:

3*8-2;
15/3+6;
0,3*8-4/2;
3/1+15/5;

A single number will also represent a numeric expression, like the number 18 from the above example.
Examples of incorrect number constructions that do not make sense:

*7-25);
16/0-;
(*-5;

Incorrect numeric examples are just a bunch of mathematical symbols and have no meaning.

How to find the value of an expression

Since such examples contain arithmetic signs, we can conclude that they allow arithmetic calculations. To calculate the signs or, in other words, to find the meaning of an expression, it is necessary to perform the appropriate arithmetic manipulations.

As an example, consider the following construction: (120-30)/3=30. The number 30 will be the value numerical expression (120-30)/3.

Instructions:

Concept of numerical equality

Numerical equality is a situation where two parts of an example are separated by the “=” sign. That is, one part is completely equal (identical) to the other, even if displayed in the form of other combinations of symbols and numbers.
For example, any construction like 2+2=4 can be called a numerical equality, since even if the parts are swapped, the meaning will not change: 4=2+2. The same goes for more complex constructions involving parentheses, division, multiplication, operations with fractions, and so on.

How to find the value of an expression correctly

To correctly find the value of the expression, you must perform calculations according to a certain order actions. This order is taught in mathematics lessons, and later in algebra classes in elementary school. It is also known as steps arithmetic operations.

Arithmetic steps:

The first stage is the addition and subtraction of numbers.
The second stage is where division and multiplication are performed.
Third stage - numbers are squared or cubed.

By observing the following rules, you can always correctly determine the meaning of an expression:

Perform actions starting from the third step, ending with the first, if there are no parentheses in the example. That is, first square or cube, then divide or multiply, and only then add and subtract.
In constructions with brackets, perform the actions in the brackets first, and then follow the order described above. If there are several brackets, also use the procedure from the first paragraph.
In examples in the form of a fraction, first find out the result in the numerator, then in the denominator, then divide the first by the second.

Finding the meaning of an expression will not be difficult if you acquire basic knowledge of elementary courses in algebra and mathematics. Guided by the information described above, you can solve any problem, even of increased complexity.

Find out the password from VK, knowing the login

An entry that consists of numbers, signs, and parentheses, and also has meaning, called a numeric expression.

For example, the following entries:

(100-32)/17,
2*4+7,
4*0.7 -3/5,
1/3 +5/7

will be numerical expressions. It should be understood that one number will also be a numerical expression. In our example, this is the number 13.

And, for example, the following entries

100 - *9,
/32)343

will not be numerical expressions, since they are meaningless and are simply a set of numbers and signs.

Numeric expression value

Since the signs in numerical expressions include signs of arithmetic operations, we can calculate the value of a numerical expression. To do this, you must follow these steps.

For example,

(100-32)/17 = 4, that is, for the expression (100-32)/17, the value of this numerical expression will be the number 4.

2*4+7=15, the number 15 will be the value of the numerical expression 2*4+7.

Often, for the sake of brevity, entries do not write the full value of a numeric expression, but simply write “the value of the expression,” while omitting the word “numeric.”

Numerical equality

If two numerical expressions are written using an equal sign, then these expressions form a numerical equality. For example, the expression 2*4+7=15 is a numerical equality.

As noted above, numeric expressions can use parentheses. As you already know, parentheses affect the order of actions.

In general, all actions are divided into several stages.

First stage actions: addition and subtraction.
Second stage operations: multiplication and division.
The actions of the third stage are squaring and cubed.

Rules for calculating the values of numeric expressions

When calculating the values of numeric expressions, the following rules should be followed.

1. If the expression does not have brackets, then you need to perform actions starting from the highest levels: third stage, second stage and first stage. If there are several actions of the same stage, then they are performed in the order in which they are written, that is, from left to right.
2. If the expression contains parentheses, then the actions in the parentheses are performed first, and only then all other actions are performed in the usual order. When performing actions in brackets, if there are several of them, you should use the order described in paragraph 1.
3. If the expression is a fraction, then the values in the numerator and denominator are first calculated, and then the numerator is divided by the denominator.
4. If the expression contains nested brackets, then actions should be performed from the inner brackets.

This article discusses how to find the values of mathematical expressions. Let's start with simple numerical expressions and then consider cases as their complexity increases. At the end we present an expression containing letter symbols, brackets, roots, special mathematical symbols, degrees, functions, etc. As per tradition, we will provide the entire theory with abundant and detailed examples.

Yandex.RTB R-A-339285-1

How to find the value of a numeric expression?

Numerical expressions, among other things, help describe the problem condition mathematical language. In general, mathematical expressions can be either very simple, consisting of a pair of numbers and arithmetic symbols, or very complex, containing functions, powers, roots, parentheses, etc. As part of a task, it is often necessary to find the meaning of a particular expression. How to do this will be discussed below.

The simplest cases

These are cases where the expression contains nothing but numbers and arithmetic operations. To successfully find the values of such expressions, you will need knowledge of the order of performing arithmetic operations without parentheses, as well as the ability to perform operations with various numbers.

If the expression contains only numbers and arithmetic signs " + " , " · " , " - " , " ÷ " , then the actions are performed from left to right in the following order: first multiplication and division, then addition and subtraction. Let's give examples.

Example 1: The value of a numeric expression

Let you need to find the values of the expression 14 - 2 · 15 ÷ 6 - 3.

Let's do the multiplication and division first. We get:

14 - 2 15 ÷ 6 - 3 = 14 - 30 ÷ 6 - 3 = 14 - 5 - 3.

Now we carry out the subtraction and get the final result:

14 - 5 - 3 = 9 - 3 = 6 .

Example 2: The value of a numeric expression

Let's calculate: 0, 5 - 2 · - 7 + 2 3 ÷ 2 3 4 · 11 12.

First we perform fraction conversion, division and multiplication:

0, 5 - 2 · - 7 + 2 3 ÷ 2 3 4 · 11 12 = 1 2 - (- 14) + 2 3 ÷ 11 4 · 11 12

1 2 - (- 14) + 2 3 ÷ 11 4 11 12 = 1 2 - (- 14) + 2 3 4 11 11 12 = 1 2 - (- 14) + 2 9.

Now let's do some addition and subtraction. Let's group the fractions and bring them to a common denominator:

1 2 - (- 14) + 2 9 = 1 2 + 14 + 2 9 = 14 + 13 18 = 14 13 18 .

The required value has been found.

Expressions with parentheses

If an expression contains parentheses, they define the order of operations in that expression. The actions in brackets are performed first, and then all the others. Let's show this with an example.

Example 3: The value of a numeric expression

Let's find the value of the expression 0.5 · (0.76 - 0.06).

The expression contains parentheses, so we first perform the subtraction operation in parentheses, and only then the multiplication.

0.5 · (0.76 - 0.06) = 0.5 · 0.7 = 0.35.

The meaning of expressions containing parentheses within parentheses is found according to the same principle.

Example 4: The value of a numeric expression

Let's calculate the value 1 + 2 · 1 + 2 · 1 + 2 · 1 - 1 4 .

We will perform actions starting from the innermost brackets, moving to the outer ones.

1 + 2 1 + 2 1 + 2 1 - 1 4 = 1 + 2 1 + 2 1 + 2 3 4

1 + 2 1 + 2 1 + 2 3 4 = 1 + 2 1 + 2 2, 5 = 1 + 2 6 = 13.

When finding the meanings of expressions with brackets, the main thing is to follow the sequence of actions.

Expressions with roots

Mathematical expressions whose values we need to find may contain root signs. Moreover, the expression itself may be under the root sign. What to do in this case? First you need to find the value of the expression under the root, and then extract the root from the number obtained as a result. If possible, it is better to get rid of roots in numerical expressions, replacing from with numeric values.

Example 5: The value of a numeric expression

Let's calculate the value of the expression with roots - 2 · 3 - 1 + 60 ÷ 4 3 + 3 · 2, 2 + 0, 1 · 0, 5.

First, we calculate the radical expressions.

2 3 - 1 + 60 ÷ 4 3 = - 6 - 1 + 15 3 = 8 3 = 2

2, 2 + 0, 1 0, 5 = 2, 2 + 0, 05 = 2, 25 = 1, 5.

Now you can calculate the value of the entire expression.

2 3 - 1 + 60 ÷ 4 3 + 3 2, 2 + 0, 1 0, 5 = 2 + 3 1, 5 = 6, 5

Often, finding the meaning of an expression with roots often requires first converting the original expression. Let's explain this with one more example.

Example 6: The value of a numeric expression

What is 3 + 1 3 - 1 - 1

As you can see, we do not have the opportunity to replace the root with an exact value, which complicates the counting process. However, in this case, you can apply the abbreviated multiplication formula.

3 + 1 3 - 1 = 3 - 1 .

Thus:

3 + 1 3 - 1 - 1 = 3 - 1 - 1 = 1 .

Expressions with powers

If an expression contains powers, their values must be calculated before proceeding with all other actions. It happens that the exponent or the base of the degree itself are expressions. In this case, the value of these expressions is first calculated, and then the value of the degree.

Example 7: The value of a numeric expression

Let's find the value of the expression 2 3 · 4 - 10 + 16 1 - 1 2 3, 5 - 2 · 1 4.

Let's start calculating in order.

2 3 4 - 10 = 2 12 - 10 = 2 2 = 4

16 · 1 - 1 2 3, 5 - 2 · 1 4 = 16 * 0, 5 3 = 16 · 1 8 = 2.

All that remains is to perform the addition operation and find out the meaning of the expression:

2 3 4 - 10 + 16 1 - 1 2 3, 5 - 2 1 4 = 4 + 2 = 6.

It is also often advisable to simplify an expression using the properties of a degree.

Example 8: The value of a numeric expression

Let's calculate the value of the following expression: 2 - 2 5 · 4 5 - 1 + 3 1 3 6 .

The exponents are again such that their exact numerical values cannot be obtained. Let's simplify the original expression to find its value.

2 - 2 5 4 5 - 1 + 3 1 3 6 = 2 - 2 5 2 2 5 - 1 + 3 1 3 6

2 - 2 5 2 2 5 - 1 + 3 1 3 6 = 2 - 2 5 2 2 5 - 2 + 3 2 = 2 2 5 - 2 - 2 5 + 3 2

2 2 5 - 2 - 2 5 + 3 2 = 2 - 2 + 3 = 1 4 + 3 = 3 1 4

Expressions with fractions

If an expression contains fractions, then when calculating such an expression, all fractions in it must be represented in the form ordinary fractions and calculate their values.

If the numerator and denominator of a fraction contain expressions, then the values of these expressions are first calculated, and the final value of the fraction itself is written down. Arithmetic operations are performed in the standard order. Let's look at the example solution.

Example 9: The value of a numeric expression

Let's find the value of the expression containing fractions: 3, 2 2 - 3 · 7 - 2 · 3 6 ÷ 1 + 2 + 3 9 - 6 ÷ 2.

As you can see, there are three fractions in the original expression. Let's first calculate their values.

3, 2 2 = 3, 2 ÷ 2 = 1, 6

7 - 2 3 6 = 7 - 6 6 = 1 6

1 + 2 + 3 9 - 6 ÷ 2 = 1 + 2 + 3 9 - 3 = 6 6 = 1.

Let's rewrite our expression and calculate its value:

1, 6 - 3 1 6 ÷ 1 = 1, 6 - 0, 5 ÷ 1 = 1, 1

Often when finding the meaning of expressions, it is convenient to reduce fractions. There is an unspoken rule: before finding its value, it is best to simplify any expression to the maximum, reducing all calculations to the simplest cases.

Example 10: The value of a numeric expression

Let's calculate the expression 2 5 - 1 - 2 5 - 7 4 - 3.

We cannot completely extract the root of five, but we can simplify the original expression through transformations.

2 5 - 1 = 2 5 + 1 5 - 1 5 + 1 = 2 5 + 1 5 - 1 = 2 5 + 2 4

The original expression takes the form:

2 5 - 1 - 2 5 - 7 4 - 3 = 2 5 + 2 4 - 2 5 - 7 4 - 3 .

Let's calculate the value of this expression:

2 5 + 2 4 - 2 5 - 7 4 - 3 = 2 5 + 2 - 2 5 + 7 4 - 3 = 9 4 - 3 = - 3 4 .

Expressions with logarithms

When logarithms are present in an expression, their value is calculated from the beginning, if possible. For example, in the expression log 2 4 + 2 · 4, you can immediately write down the value of this logarithm instead of log 2 4, and then perform all the actions. We get: log 2 4 + 2 4 = 2 + 2 4 = 2 + 8 = 10.

Numerical expressions can also be found under the logarithm sign itself and at its base. In this case, the first thing to do is find their meanings. Let's take the expression log 5 - 6 ÷ 3 5 2 + 2 + 7. We have:

log 5 - 6 ÷ 3 5 2 + 2 + 7 = log 3 27 + 7 = 3 + 7 = 10.

If it is impossible to calculate the exact value of the logarithm, simplifying the expression helps to find its value.

Example 11: The value of a numeric expression

Let's find the value of the expression log 2 log 2 256 + log 6 2 + log 6 3 + log 5 729 log 0, 2 27.

log 2 log 2 256 = log 2 8 = 3 .

By the property of logarithms:

log 6 2 + log 6 3 = log 6 (2 3) = log 6 6 = 1.

Using the properties of logarithms again, for the last fraction in the expression we get:

log 5 729 log 0, 2 27 = log 5 729 log 1 5 27 = log 5 729 - log 5 27 = - log 27 729 = - log 27 27 2 = - 2.

Now you can proceed to calculating the value of the original expression.

log 2 log 2 256 + log 6 2 + log 6 3 + log 5 729 log 0, 2 27 = 3 + 1 + - 2 = 2.

Expressions with trigonometric functions

It happens that the expression contains the trigonometric functions of sine, cosine, tangent and cotangent, as well as their inverse functions. The value is calculated from before all other arithmetic operations are performed. Otherwise, the expression is simplified.

Example 12: The value of a numeric expression

Find the value of the expression: t g 2 4 π 3 - sin - 5 π 2 + cosπ.

First, we calculate the values of the trigonometric functions included in the expression.

sin - 5 π 2 = - 1

We substitute the values into the expression and calculate its value:

t g 2 4 π 3 - sin - 5 π 2 + cosπ = 3 2 - (- 1) + (- 1) = 3 + 1 - 1 = 3.

The expression value has been found.

Often in order to find the meaning of an expression with trigonometric functions, it must first be converted. Let's explain with an example.

Example 13: The value of a numeric expression

We need to find the value of the expression cos 2 π 8 - sin 2 π 8 cos 5 π 36 cos π 9 - sin 5 π 36 sin π 9 - 1.

For conversion we will use trigonometric formulas cosine of the double angle and cosine of the sum.

cos 2 π 8 - sin 2 π 8 cos 5 π 36 cos π 9 - sin 5 π 36 sin π 9 - 1 = cos 2 π 8 cos 5 π 36 + π 9 - 1 = cos π 4 cos π 4 - 1 = 1 - 1 = 0.

General case of a numeric expression

In general, a trigonometric expression can contain all the elements described above: brackets, powers, roots, logarithms, functions. Let's formulate general rule finding the meanings of such expressions.

How to find the value of an expression

Roots, powers, logarithms, etc. are replaced by their values.
The actions in parentheses are performed.
The remaining actions are performed in order from left to right. First - multiplication and division, then - addition and subtraction.

Let's look at an example.

Example 14: The value of a numeric expression

Let's calculate the value of the expression - 2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 ln e 2 + 1 + 3 9.

The expression is quite complex and cumbersome. It was not by chance that we chose just such an example, having tried to fit into it all the cases described above. How to find the meaning of such an expression?

It is known that when calculating the value of a complex fractional form, the values of the numerator and denominator of the fraction are first found separately, respectively. We will sequentially transform and simplify this expression.

First of all, let's calculate the value of the radical expression 2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3. To do this, you need to find the value of the sine and the expression that is the argument of the trigonometric function.

π 6 + 2 2 π 5 + 3 π 5 = π 6 + 2 2 π + 3 π 5 = π 6 + 2 5 π 5 = π 6 + 2 π

Now you can find out the value of the sine:

sin π 6 + 2 2 π 5 + 3 π 5 = sin π 6 + 2 π = sin π 6 = 1 2.

We calculate the value of the radical expression:

2 sin π 6 + 2 2 π 5 + 3 π 5 + 3 = 2 1 2 + 3 = 4

2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 = 4 = 2.

With the denominator of the fraction everything is simpler:

Now we can write the value of the entire fraction:

2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 ln e 2 = 2 2 = 1 .

Taking this into account, we write the entire expression:

1 + 1 + 3 9 = - 1 + 1 + 3 3 = - 1 + 1 + 27 = 27 .

Final result:

2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 ln e 2 + 1 + 3 9 = 27.

In this case we were able to calculate exact values roots, logarithms, sines, etc. If this is not possible, you can try to get rid of them through mathematical transformations.

Calculating expression values using rational methods

Numeric values must be calculated consistently and accurately. This process can be rationalized and accelerated using various properties of operations with numbers. For example, it is known that a product is equal to zero if at least one of the factors is equal to zero. Taking this property into account, we can immediately say that the expression 2 386 + 5 + 589 4 1 - sin 3 π 4 0 is equal to zero. At the same time, it is not at all necessary to perform the actions in the order described in the article above.

It is also convenient to use the subtraction property equal numbers. Without performing any actions, you can order that the value of the expression 56 + 8 - 3, 789 ln e 2 - 56 + 8 - 3, 789 ln e 2 is also zero.

Another technique to speed up the process is the use of identity transformations such as grouping terms and factors and placing the common factor out of brackets. Rational approach to calculating expressions with fractions - shortcut identical expressions in the numerator and denominator.

For example, take the expression 2 3 - 1 5 + 3 289 3 4 3 2 3 - 1 5 + 3 289 3 4. Without performing the operations in parentheses, but by reducing the fraction, we can say that the value of the expression is 1 3 .

Finding the values of expressions with variables

Meaning literal expression and expressions with variables are found for specific given values of letters and variables.

Finding the values of expressions with variables

To find the value of a literal expression and an expression with variables, you need to substitute the given values of letters and variables into the original expression, and then calculate the value of the resulting numeric expression.

Example 15: The value of an expression with variables

Calculate the value of the expression 0, 5 x - y given x = 2, 4 and y = 5.

We substitute the values of the variables into the expression and calculate:

0.5 x - y = 0.5 2.4 - 5 = 1.2 - 5 = - 3.8.

Sometimes you can transform an expression so that you get its value regardless of the values of the letters and variables included in it. To do this, you need to get rid of letters and variables in the expression, if possible, using identical transformations, properties of arithmetic operations and all possible other methods.

For example, the expression x + 3 - x obviously has the value 3, and to calculate this value it is not necessary to know the value of the variable x. The value of this expression is equal to three for all values of the variable x from its range of permissible values.

Another example. The value of the expression x x is equal to one for all positive x's.

If you notice an error in the text, please highlight it and press Ctrl+Enter

You, as parents, in the process of educating your child, will more than once encounter the need for help in solving homework problems in mathematics, algebra and geometry. And one of the basic skills that you need to learn is how to find the meaning of an expression. Many people are at a dead end, because how many years have passed since we studied in grades 3-5? Much has already been forgotten, and some have not been learned. The rules of mathematical operations themselves are simple and you can easily remember them. Let's start with the very basics of what a mathematical expression is.

Expression Definition

A mathematical expression is a collection of numbers, action signs (=, +, -, *, /), brackets, and variables. Briefly, this is a formula whose value will need to be found. Such formulas are found in mathematics courses since school, and then haunt students who have chosen specialties related to exact sciences. Mathematical expressions are divided into trigonometric, algebraic, and so on; let’s not get into the thicket.

Do any calculations first on a draft, and then rewrite them in workbook. This way you will avoid unnecessary crossings and dirt;
Recalculate total quantity mathematical operations that will need to be performed in the expression. Please note that according to the rules, the operations in brackets are performed first, then division and multiplication, and at the very end subtraction and addition. We recommend highlighting all the actions in pencil and putting numbers above the actions in the order in which they were performed. In this case, it will be easier for both you and your child to navigate;
Start making calculations strictly following the order of actions. Let the child, if the calculation is simple, try to perform it in his head, but if it is difficult, then write with a pencil the number corresponding serial number expressions and perform calculations in writing under the formula;
As a rule, finding the value of a simple expression is not difficult if all calculations are carried out in accordance with the rules and in the right order. Most people encounter a problem precisely at this stage of finding the meaning of an expression, so be careful and do not make mistakes;
Ban the calculator. The mathematical formulas and problems themselves may not be useful in your child’s life, but that is not the purpose of studying the subject. The main thing is development logical thinking. If you use calculators, the meaning of everything will be lost;
Your task as a parent is not to solve problems for your child, but to help him in this, to guide him. Let him make all the calculations himself, and you make sure that he doesn’t make mistakes, explain why he needs to do it this way and not otherwise.
Once the answer to the expression has been found, write it down after the “=” sign;
Open the last page of your math textbook. Usually, there are answers for every exercise in the book. It doesn’t hurt to check whether everything has been calculated correctly.

Finding the meaning of an expression is, on the one hand, a simple procedure; the main thing is to remember the basic rules that we went through in school course mathematics. However, on the other hand, when you need to help your child cope with formulas and solve problems, the issue becomes more complicated. After all, you are now not a student, but a teacher, and the education of the future Einstein rests on your shoulders.

We hope that our article helped you find the answer to the question of how to find the meaning of an expression, and you can easily figure out any formula!

I. Expressions in which numbers, arithmetic symbols and parentheses can be used along with letters are called algebraic expressions.

Examples of algebraic expressions:

2m -n; 3 · (2a + b); 0.24x; 0.3a -b · (4a + 2b); a 2 – 2ab;

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

II. If in an algebraic expression the letters (variables) are replaced by their values and the specified actions are performed, then the resulting number is called the value of the algebraic expression.

Examples. Find the meaning of the expression:

1) a + 2b -c with a = -2; b = 10; c = -3.5.

2) |x| + |y| -|z| at x = -8; y = -5; z = 6.

Solution.

1) a + 2b -c with a = -2; b = 10; c = -3.5. Instead of variables, let's substitute their values. We get:

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

2) |x| + |y| -|z| at x = -8; y = -5; z = 6. Substitute the indicated values. Remember that the module negative number is equal to its opposite number, and the module positive number equal to this number itself. We get:

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

III. The values of the letter (variable) for which the algebraic expression makes sense are called the permissible values of the letter (variable).

Examples. For what values of the variable does the expression make no sense?

Solution. We know that you cannot divide by zero, therefore, each of these expressions will not make sense given the value of the letter (variable) that turns the denominator of the fraction to zero!

In example 1) this value is a = 0. Indeed, if you substitute 0 instead of a, then you will need to divide the number 6 by 0, but this cannot be done. Answer: expression 1) does not make sense when a = 0.

In example 2) the denominator of x is 4 = 0 at x = 4, therefore, this value x = 4 cannot be taken. Answer: expression 2) does not make sense when x = 4.

In example 3) the denominator is x + 2 = 0 when x = -2. Answer: expression 3) does not make sense when x = -2.

In example 4) the denominator is 5 -|x| = 0 for |x| = 5. And since |5| = 5 and |-5| = 5, then you cannot take x = 5 and x = -5. Answer: expression 4) does not make sense at x = -5 and at x = 5.
IV. Two expressions are said to be identically equal if for any acceptable values variables, the corresponding values of these expressions are equal.

Example: 5 (a – b) and 5a – 5b are also equal, since the equality 5 (a – b) = 5a – 5b will be true for any values of a and b. The equality 5 (a – b) = 5a – 5b is an identity.

Identity is an equality that is valid for all permissible values of the variables included in it. Examples of identities already known to you are, for example, the properties of addition and multiplication, and the distributive property.

Replacing one expression with another identically equal expression is called an identity transformation or simply a transformation of an expression. Identical transformations of expressions with variables are performed based on the properties of operations on numbers.

Examples.

a) convert the expression to identically equal using the distributive property of multiplication:

1) 10·(1.2x + 2.3y); 2) 1.5·(a -2b + 4c); 3) a·(6m -2n + k).

Solution. Let us recall the distributive property (law) of multiplication:

(a+b)c=ac+bc(distributive law of multiplication relative to addition: in order to multiply the sum of two numbers by a third number, you can multiply each term by this number and add the resulting results).
(a-b) c=a c-b c(distributive law of multiplication relative to subtraction: in order to multiply the difference of two numbers by a third number, you can multiply the minuend and subtract by this number separately and subtract the second from the first result).

1) 10·(1.2x + 2.3y) = 10 · 1.2x + 10 · 2.3y = 12x + 23y.

2) 1.5·(a -2b + 4c) = 1.5a -3b + 6c.

3) a·(6m -2n + k) = 6am -2an +ak.

b) transform the expression into identically equal, using the commutative and associative properties (laws) of addition:

4) x + 4.5 +2x + 6.5; 5) (3a + 2.1) + 7.8; 6) 5.4s -3 -2.5 -2.3s.

Solution. Let's apply the laws (properties) of addition:

a+b=b+a(commutative: rearranging the terms does not change the sum).
(a+b)+c=a+(b+c)(combinative: in order to add a third number to the sum of two terms, you can add the sum of the second and third to the first number).

4) x + 4.5 +2x + 6.5 = (x + 2x) + (4.5 + 6.5) = 3x + 11.

5) (3a + 2.1) + 7.8 = 3a + (2.1 + 7.8) = 3a + 9.9.

6) 6) 5.4s -3 -2.5 -2.3s = (5.4s -2.3s) + (-3 -2.5) = 3.1s -5.5.

V) Convert the expression to identically equal using the commutative and associative properties (laws) of multiplication:

7) 4 · X · (-2,5); 8) -3,5 · 2у · (-1); 9) 3a · (-3) · 2s.

Solution. Let's apply the laws (properties) of multiplication:

a·b=b·a(commutative: rearranging the factors does not change the product).
(a b) c=a (b c)(combinative: to multiply the product of two numbers by a third number, you can multiply the first number by the product of the second and third).