When studying the topic of numerical, literal expressions and expressions with variables, it is necessary to pay attention to the concept expression value. In this article, we will answer the question, what is the value of a numeric expression, and what is called the value of a literal expression and an expression with variables with the selected values ​​of the variables. To clarify these definitions, we give examples.

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What is the value of a numeric expression?

Acquaintance with numerical expressions begins almost from the first lessons of mathematics at school. Almost immediately, the concept of “value of a numerical expression” is introduced. It refers to expressions made up of numbers connected by arithmetic signs (+, −, ·, :). Let us give an appropriate definition.

Definition.

The value of a numeric expression- this is the number that is obtained after performing all the actions in the original numeric expression.

For example, consider the numeric expression 1+2 . After executing , we get the number 3 , it is the value of the numerical expression 1+2 .

Often in the phrase “value of a numerical expression”, the word “numerical” is omitted, and they simply say “value of the expression”, since it is still clear which expression is meant.

The above definition of the meaning of an expression also applies to numerical expressions of a more complex form, which are studied in high school. Here it should be noted that one may encounter numerical expressions, the values ​​of which cannot be specified. This is due to the fact that in some expressions it is impossible to perform the recorded actions. For example, therefore we cannot specify the value of the expression 3:(2−2) . Such numerical expressions are called expressions that don't make sense.

Often in practice, it is not so much the numerical expression that is of interest as its value. That is, the task arises, which consists in determining the value of this expression. In this case, they usually say that you need to find the value of the expression. In this article, the process of finding the value of numerical expressions of various types is analyzed in detail, and a lot of examples with detailed descriptions of solutions are considered.

Meaning of literal and variable expressions

In addition to numerical expressions, they study literal expressions, that is, expressions in which, along with numbers, one or more letters are present. Letters in a literal expression can stand for different numbers, and if the letters are replaced by these numbers, then the literal expression becomes a numeric one.

Definition.

The numbers that replace letters in a literal expression are called the meanings of these letters, and the value of the resulting numerical expression is called the value of the literal expression given the values ​​of the letters.

So, for literal expressions, one speaks not just about the meaning of the literal expression, but about the meaning of the literal expression for the given (given, indicated, etc.) values ​​of the letters.

Let's take an example. Let's take the literal expression 2·a+b . Let the values ​​of the letters a and b be given, for example, a=1 and b=6 . Replacing the letters in the original expression with their values, we get a numerical expression of the form 2 1+6 , its value is 8 . Thus, the number 8 is the value of the literal expression 2·a+b given the values ​​of the letters a=1 and b=6 . If other letter values ​​were given, then we would get the value of the literal expression for those letter values. For example, with a=5 and b=1 we have the value 2 5+1=11 .

In high school, when studying algebra, letters in literal expressions are allowed to take on different meanings, such letters are called variables, and literal expressions are called expressions with variables. For these expressions, the concept of the value of an expression with variables is introduced for the chosen values ​​of the variables. Let's figure out what it is.

Definition.

The value of an expression with variables for the selected values ​​of the variables the value of a numeric expression is called, which is obtained after substituting the selected values ​​of the variables into the original expression.

Let us explain the sounded definition with an example. Consider an expression with variables x and y of the form 3·x·y+y . Let's take x=2 and y=4 , substitute these variable values ​​into the original expression, we get the numerical expression 3 2 4+4 . Let's calculate the value of this expression: 3 2 4+4=24+4=28 . The found value 28 is the value of the original expression with the variables 3·x·y+y with the selected values ​​of the variables x=2 and y=4 .

If you choose other values ​​of variables, for example, x=5 and y=0 , then these selected values ​​of variables will correspond to the value of the expression with variables equal to 3 5 0+0=0 .

It can be noted that sometimes equal values ​​of the expression can be obtained for different chosen values ​​of variables. For example, for x=9 and y=1, the value of the expression 3 x y+y is 28 (because 3 9 1+1=27+1=28 ), and above we showed that the same value is expression with variables has at x=2 and y=4 .

Variable values ​​can be selected from their respective ranges of acceptable values. Otherwise, substituting the values ​​of these variables into the original expression will result in a numerical expression that does not make sense. For example, if you choose x=0 , and substitute that value into the expression 1/x , you get the numeric expression 1/0 , which doesn't make sense because division by zero is undefined.

It only remains to add that there are expressions with variables whose values ​​do not depend on the values ​​of their constituent variables. For example, the value of an expression with a variable x of the form 2+x−x does not depend on the value of this variable, it is equal to 2 for any chosen value of the variable x from its range of valid values, which in this case is the set of all real numbers.

Bibliography.

• Mathematics: studies. for 5 cells. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 p.: ill. ISBN 5-346-00699-0.
• Algebra: textbook for 7 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 17th ed. - M. : Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
• Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.

Formula

Addition, subtraction, multiplication, division - arithmetic operations (or arithmetic operations). These arithmetic operations correspond to the signs of arithmetic operations:

+ (read " plus") - the sign of the addition operation,

- (read " minus") - the sign of the subtraction operation,

∙ (read " multiply") - the sign of the multiplication operation,

: (read " divide") is the sign of the division operation.

A record consisting of numbers interconnected by signs of arithmetic operations is called numerical expression. Parentheses can also be present in a numeric expression. For example, entry 1290 : 2 - (3 + 20 ∙ 15) is a numeric expression.

The result of performing operations on numbers in a numerical expression is called the value of a numeric expression. Performing these actions is called calculating the value of a numeric expression. Before writing the value of a numeric expression, put equal sign"=". Table 1 shows examples of numeric expressions and their meanings.

A record consisting of numbers and small letters of the Latin alphabet, interconnected by signs of arithmetic operations is called literal expression. This entry may contain parentheses. For example, the entry a +b - 3 ∙c is a literal expression. Instead of letters in a literal expression, you can substitute various numbers. In this case, the meaning of the letters can change, so the letters in the literal expression are also called variables.

Substituting numbers instead of letters into the literal expression and calculating the value of the resulting numerical expression, they find the value of a literal expression given the values ​​of the letters(for the given values ​​of the variables). Table 2 shows examples of literal expressions.

A literal expression may not have a value if, by substituting the values ​​of the letters, a numeric expression is obtained whose value for natural numbers cannot be found. Such a numerical expression is called incorrect for natural numbers. They also say that the meaning of such an expression " undefined" for natural numbers, and the expression itself "doesn't make sense". For example, the literal expression a-b does not matter for a = 10 and b = 17. Indeed, for natural numbers, the minuend cannot be less than the subtrahend. For example, having only 10 apples (a = 10), you cannot give away 17 of them (b = 17)!

Table 2 (column 2) shows an example of a literal expression. By analogy, fill in the table completely.

For natural numbers, the expression 10 -17 wrong (doesn't make sense), i.e. the difference 10 -17 cannot be expressed as a natural number. Another example: you cannot divide by zero, so for any natural number b, the quotient b:0 undefined.

Mathematical laws, properties, some rules and ratios are often written in literal form (i.e. in the form of a literal expression). In these cases, the literal expression is called formula. For example, if the sides of a heptagon are equal a,b,c,d,e,f,g, then the formula (literal expression) for calculating its perimeter p looks like:

p=a +b+c+d+e +f +g

For a = 1, b = 2, c = 4, d = 5, e = 5, f = 7, g = 9, the perimeter of the heptagon is p = a + b + c + d + e + f + g = 1 + 2 + 4 + 5 +5 + 7 + 9 = 33.

For a = 12, b = 5, c = 20, d = 35, e = 4, f = 40, g = 18, the perimeter of another heptagon is p = a + b + c + d + e + f + g = 12 + 5 + 20 + 35 + 4 + 40 + 18 = 134.

Block 1. Dictionary

Make a dictionary of new terms and definitions from the paragraph. To do this, in the empty cells, enter the words from the list of terms below. In the table (at the end of the block), indicate the numbers of terms in accordance with the numbers of the frames. It is recommended to carefully review the paragraph before filling in the cells of the dictionary.

1. Operations: addition, subtraction, multiplication, division.

2. Signs "+" (plus), "-" (minus), "∙" (multiply, " : " (divide).

3. A record consisting of numbers that are interconnected by signs of arithmetic operations and in which brackets may also be present.

4. The result of performing operations on numbers in numerical terms.

5. The sign before the value of a numeric expression.

6. An entry consisting of numbers and small letters of the Latin alphabet, interconnected by signs of arithmetic operations (brackets may also be present).

7. The common name of the letters in the literal expression.

8. The value of a numeric expression, which is obtained by substituting variables into a literal expression.

9. Numeric expression whose value for natural numbers cannot be found.

10. Numeric expression whose value for natural numbers can be found.

11. Mathematical laws, properties, some rules and ratios written in literal form.

12. An alphabet whose small letters are used to write literal expressions.

Block 2. Match

Match the task in the left column with the solution in the right. Write down the answer in the form: 1a, 2d, 3b ...

Block 3. Facet test. Numeric and alphabetic expressions

Faceted tests replace collections of problems in mathematics, but compare favorably with them in that they can be solved on a computer, check solutions and immediately find out the result of the work. This test contains 70 tasks. But you can solve problems by choice, for this there is an evaluation table, which lists simple tasks and more difficult ones. Below is a test.

1. Given a triangle with sides c,d,m, expressed in cm
2. Given a quadrilateral with sides b,c,d,m expressed in m
3. The speed of the car in km/h is b, travel time in hours is d
4. Distance traveled by a tourist m hours, is with km
5. The distance traveled by a tourist moving at a speed m km/h is b km
6. The sum of two numbers is greater than the second number by 15
7. The difference is less than the reduced by 7
8. A passenger liner has two decks with the same number of passenger seats. In each of the deck rows m seats, rows on deck on n more than seats in a row
9. Petya is m years old Masha is n years old, and Katya is k years younger than Petya and Masha together
10. m=8, n=10, k=5
11. m=6, n=8, k=15
12. t=121, x=1458

1. The value of this expression
2. The literal expression for the perimeter is
3. Perimeter expressed in centimeters
4. Formula for the distance s traveled by the car
5. Velocity formula v, tourist movements
6. Time formula t, tourist movements
7. Distance traveled by car in kilometers
8. Tourist speed in kilometers per hour
9. Travel time in hours
10. The first number is...
11. Subtracted equals….
12. The expression for the largest number of passengers that the liner can carry in k flights
13. The largest number of passengers that an airliner can carry in k flights
14. Letter expression for Katya's age
15. Katya's age
16. The coordinate of point B, if the coordinate of point C is t
17. The coordinate of point D, if the coordinate of point C is t
18. The coordinate of point A, if the coordinate of point C is t
19. The length of the segment BD on the number line
20. The length of the segment CA on the number line
21. The length of the segment DA on the number line

A numeric expression is a record of numbers in conjunction with arithmetic operations and brackets. When variables are used in an expression together with numbers and the whole expression is composed with meaning, then it is called an algebraic (literal) expression. If the expression contains direct, derivative, inverse and other trigonometric functions, then the expression is called trigonometric. A large number of examples and tasks using various expressions are detailed in the school mathematics course.

The main things to remember:

1. The value of a numeric expression will be the number obtained by performing arithmetic operations in this expression. The main thing is to consistently perform arithmetic operations. For simplicity of the whole operation, the steps can be numbered. If the expression contains brackets, then first of all we perform the action corresponding to the character in brackets. Exponentiation will be the next step. Next in priority, we perform multiplication or division, and only at the very end, addition and subtraction.

Now let's find the value of the numerical expression 5+20*(60-45). Let's get rid of the parentheses first. Performing the action, we get 60-45=15. Now we have 5+20*15. The next action is multiplication 20*15=300. And the last action will be addition, we perform it and get the final result 5 + 300 = 305.

2. At a known angle? When working with trigonometric expressions, you will need knowledge of basic trigonometric formulas that will help simplify the expression. Let's find the value of the expression cos 12? cos 18? - sin 12? sin 18?. To simplify this expression, we use the formula cos (? +?) = cos? cos? - sin? sin?, then we get cos 12? cos 18? - sin 12? sin 18?= cos(12? +18?)= cos30? =v3?2.

3. Expressions with variables. It must be remembered that the value of an algebraic expression directly depends on the variable. Variables can be denoted by letters of the Greek or Latin alphabet. When we have the given parameters of an algebraic expression, we first need to simplify it. After that, it is necessary to substitute the given variables and perform arithmetic operations. As a result, with the given variables, we will get a number, which will be the value of the algebraic expression. Consider an example where you need to find the value of the expression 3(a+y)+2(3a+2y) with a=4 and y=5. Simplify this expression and get 3a+3y+6a+4y=9a+7y. Now you need to substitute the value of the variables and calculate, the result obtained will be the value of the expression. So we have 9a+7y with a=4 and y=5 we get 36+35=71. Note that algebraic expressions do not always make sense. For example, the expression 15:(b-4) makes sense for any b other than b =4.

This article discusses how to find the values ​​of mathematical expressions. Let's start with simple numerical expressions and then we will consider cases as their complexity increases. At the end, we give an expression containing letter designations, brackets, roots, special mathematical signs, degrees, functions, etc. The whole theory, according to tradition, will be provided with abundant and detailed examples.

Yandex.RTB R-A-339285-1

How to find the value of a numeric expression?

Numeric expressions, among other things, help to describe the condition of the problem in mathematical language. In general, mathematical expressions can be either very simple, consisting of a pair of numbers and arithmetic signs, or very complex, containing functions, degrees, roots, brackets, etc. As part of the task, it is often necessary to find the value of an expression. How to do this will be discussed below.

The simplest cases

These are cases where the expression contains nothing but numbers and arithmetic. To successfully find the values ​​of such expressions, you will need knowledge of the order in which arithmetic operations are performed without brackets, as well as the ability to perform operations with different numbers.

If the expression contains only numbers and arithmetic signs " + " , " · " , " - " , " ÷ " , then operations 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 it be necessary 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 subtract 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 the conversion of fractions, 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 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 desired value is found.

Expressions with brackets

If an expression contains brackets, then they determine the order of actions in this expression. First, the actions in brackets are performed, and then all the rest. Let's show this with an example.

Example 3. The value of a numeric expression

Find the value of the expression 0 . 5 · (0 . 76 - 0 . 06) .

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

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

The value of expressions containing brackets in brackets 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 .

In finding the values ​​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 can be under the sign of the root. How to be in that case? First you need to find the value of the expression under the root, and then extract the root from the resulting number. If possible, it is better to get rid of roots in numerical expressions, replacing from with numerical 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 we 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, to find the value of an expression with roots, it is often necessary to first transform the original expression. Let's explain this with another 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 ability 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 the expression contains powers, their values ​​must be calculated before proceeding with all other actions. It happens that the exponent itself or the base of the degree are expressions. In this case, the value of these expressions is calculated first, and then the value of the degree.

Example 7. The value of a numeric expression

Find the value of the expression 2 3 4 - 10 + 16 1 - 1 2 3 , 5 - 2 1 4 .

We begin to calculate 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.

It remains only to carry out the addition operation and find out the value 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 the expression using the properties of the 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. 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 as ordinary fractions and their values ​​​​calculated.

If there are expressions in the numerator and denominator of the fraction, then the values ​​of these expressions are first calculated, and the final value of the fraction itself is recorded. Arithmetic operations are performed in the standard order. Let's consider an 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 us 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 values ​​of expressions, it is convenient to reduce fractions. There is an unspoken rule: before finding its value, any expression is best simplified 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, if possible, is calculated from the very beginning. For example, in the expression log 2 4 + 2 4, you can immediately write 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 .

Numeric expressions can also be found under the sign of the logarithm and at its base. In this case, the first step is to find their values. 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

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 .

According to the property of logarithms:

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

Again applying the properties of logarithms, 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 the calculation of 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 in the expression there are trigonometric functions of sine, cosine, tangent and cotangent, as well as functions that are inverse to them. From the value are calculated 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 \u003d - 1

Substitute the values ​​in the expression and calculate its value:

t g 2 4 π 3 - sin - 5 π 2 + cosπ \u003d 3 2 - (- 1) + (- 1) \u003d 3 + 1 - 1 \u003d 3.

The value of the expression is found.

Often, in order to find the value 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

It is necessary 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 the transformation, we will use the trigonometric formulas for the cosine of the double angle and the 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 numeric expression

In the general case, a trigonometric expression can contain all the elements described above: brackets, degrees, roots, logarithms, functions. Let us formulate a general rule for finding the values ​​of such expressions.

How to find the value of an expression

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

Let's take an example.

Example 14. The value of a numeric expression

Let's calculate what the value of the expression is - 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 is not by chance that we chose just such an example, trying to fit into it all the cases described above. How to find the value of such an expression?

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

First of all, we 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 a fraction, everything is easier:

Now we can write down the value of the whole fraction:

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

With this in mind, 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 ​​for roots, logarithms, sines, and so on. If this is not possible, you can try to get rid of them by mathematical transformations.

Computing Expressions in Rational Ways

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

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

Another technique that allows you to speed up the process is the use of identical transformations such as grouping terms and factors and taking the common factor out of brackets. A rational approach to calculating expressions with fractions is to reduce the same expressions in the numerator and denominator.

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

Finding the values ​​of expressions with variables

The value of a literal expression and an expression with variables is 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 it is possible to transform an expression in such a way as to obtain its value regardless of the values ​​of the letters and variables included in it. To do this, it is necessary 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 it is not necessary to know the value of x to calculate this value. The value of this expression is equal to three for all values ​​of the variable x from its range of valid values.

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

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You, as parents, in the process of teaching your child, will often face 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 value of an expression. Many come to a standstill, because how many years have passed since we were in grades 3-5? Much has already been forgotten, but something has 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

Mathematical expression - a set of numbers, action signs (=, +, -, *, /), brackets, variables. Briefly, this is a formula whose value will need to be found. Such formulas are just found in the course of mathematics since school, and then they persecute students who have chosen specialties related to the exact sciences. Mathematical expressions are divided into trigonometric, algebraic and so on, we will not run into the very "wilds".

1. Do any calculations first on a draft, and then rewrite it in a workbook. Thus, you will avoid unnecessary strikethroughs and dirt;
2. Recalculate the total number of mathematical operations that will need to be performed in the expression. Please note that according to the rules, operations in brackets are performed first, then division and multiplication, and at the very end, subtraction and addition. We recommend that you highlight all the actions with a pencil and put numbers above the actions in the order in which they are performed. In this case, it will be easier for you and the child to navigate;
3. Start making calculations strictly adhering to the order in which the actions are performed. Let the child, if the calculation is simple, try to do it in his mind, but if it is difficult, then put in a pencil the number corresponding to the ordinal number of the expression and do the calculation in writing under the formula;
4. As a rule, finding the value of a simple expression is not difficult if all the calculations are performed in accordance with the rules and the correct order. Most are faced with a problem at this stage of finding the value of the expression, so be careful and do not make mistakes;
5. Ban the calculator. Mathematical formulas and tasks themselves may not be useful to your child, but this is not the purpose of studying the subject. The main thing is the development of logical thinking. If you use calculators, then the meaning of everything will be lost;
6. Your task as a parent is not to solve problems for the child, but to help him in this, to guide him. Let him do all the calculations himself, and you make sure that he does not make mistakes, explain why you need to do it this way and not otherwise.
7. After the answer to the expression is found, write it down after the "=" sign;
8. Open the last page of your math textbook. Usually, there are answers for every exercise in the book. It does not interfere with checking whether everything is calculated correctly.

Finding the value 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 the school mathematics course. However, on the other hand, when you need to help your baby cope with formulas and problem solving, the issue becomes more complicated. After all, you are now not a student, but a teacher, and the upbringing of the future Einstein lies on your shoulders.

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