Expansion of brackets: rules, examples, solutions. Numeric expressions

The main function of brackets is to change the order of actions when calculating values. for instance, in the numerical expression \ (5 3 + 7 \), multiplication will be calculated first, and then addition: \ (5 3 + 7 = 15 + 7 = 22 \). But in the expression \ (5

Example. Expand the bracket: \ (- (4m + 3) \).
Solution : \ (- (4m + 3) = - 4m-3 \).

Example. Expand the parenthesis and give similar terms \ (5- (3x + 2) + (2 + 3x) \).
Solution : \ (5- (3x + 2) + (2 + 3x) = 5-3x-2 + 2 + 3x = 5 \).

Example. Expand the brackets \ (5 (3-x) \).
Solution : In the bracket we have \ (3 \) and \ (- x \), and in front of the bracket there is a five. Hence, each member of the bracket is multiplied by \ (5 \) - I remind you that the multiplication sign between a number and a parenthesis is not written in mathematics to reduce the size of records.

Example. Expand the brackets \ (- 2 (-3x + 5) \).
Solution : As in the previous example, \ (- 3x \) and \ (5 \) are multiplied by \ (- 2 \).

Example. Simplify expression: \ (5 (x + y) -2 (x-y) \).
Solution : \ (5 (x + y) -2 (x-y) = 5x + 5y-2x + 2y = 3x + 7y \).

It remains to consider the last situation.

When multiplying a parenthesis by a parenthesis, each member of the first parenthesis is multiplied with each member of the second:

\ ((c + d) (a-b) = c (a-b) + d (a-b) = ca-cb + da-db \)

Example. Expand the brackets \ ((2-x) (3x-1) \).
Solution : We have a product of parentheses and it can be expanded immediately using the formula above. But in order not to get confused, let's do everything in steps.
Step 1. Remove the first bracket - we multiply each of its members by the second bracket:

Step 2. Expand the product of the parenthesis by the factor as described above:
- first the first ...

Then the second.

Step 3. Now we multiply and give similar terms:

It is not at all necessary to describe all the transformations in such detail, you can immediately multiply. But if you are just learning to open parentheses - write in detail, there will be less chance of making a mistake.

A note to the entire section. In fact, you do not need to memorize all four rules, it is enough to remember only one, this is: \ (c (a-b) = ca-cb \). Why? Because if you substitute one instead of c in it, you get the rule \ ((a-b) = a-b \). And if we substitute minus one, we get the rule \ (- (a-b) = - a + b \). Well, if instead of c you substitute another parenthesis, you can get the last rule.

Parenthesis in parenthesis

Sometimes in practice there are problems with parentheses nested inside other parentheses. Here is an example of such a task: simplify the expression \ (7x + 2 (5- (3x + y)) \).

To successfully solve such tasks, you need:
- carefully understand the nesting of brackets - which one is in which;
- expand parentheses sequentially, starting, for example, from the innermost one.

In this case, it is important when opening one of the brackets do not touch the rest of the expression by simply rewriting it as it is.
Let's take the above task as an example.

Example. Expand the brackets and give similar terms \ (7x + 2 (5- (3x + y)) \).
Solution:

Example. Expand the parentheses and give similar terms \ (- (x + 3 (2x-1 + (x-5))) \).
Solution :

\ (- (x + 3 (2x-1 \) \ (+ (x-5) \) \ ()) \)		Here is a triple nesting of parentheses. We start with the innermost one (highlighted in green). There is a plus in front of the bracket, so it can be easily removed.
\ (- (x + 3 (2x-1 \) \ (+ x-5 \) \ ()) \)		Now you need to expand the second parenthesis, the intermediate one. But before that we simplify the expression with a ghost similar to the terms in this second parenthesis.
\ (= - (x \) \ (+ 3 (3x-6) \) \ () = \)		Now we open the second parenthesis (highlighted in blue). There is a factor in front of the parenthesis - so each term in the parenthesis is multiplied by it.
\ (= - (x \) \ (+ 9x-18 \) \ () = \)
		And we open the last parenthesis. Before the parenthesis there is a minus - therefore all signs are reversed.

Opening parentheses is a basic skill in mathematics. Without this skill, it is impossible to have a grade above three in the 8th and 9th grade. Therefore, I recommend that you understand this topic well.

Expanding parentheses is a type of expression conversion. In this section, we will describe the rules for expanding parentheses, and also consider the most common examples of tasks.

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What is called expanding parentheses?

Brackets are used to indicate the order in which actions are performed in numeric, literal, and variable expressions. It is convenient to pass from an expression with parentheses to an identically equal expression without parentheses. For example, replace the expression 2 (3 + 4) with an expression of the form 2 3 + 2 4 without brackets. This technique is called parenthesis expansion.

Definition 1

Bracket expansion is understood as a technique for getting rid of brackets and is usually considered in relation to expressions that may contain:

• signs "+" or "-" in front of brackets, which enclose sums or differences;
• product of a number, letter or several letters and the sum or difference, which is placed in brackets.

This is how we are used to considering the process of opening parentheses in the course of the school curriculum. However, no one prevents us from looking at this action more broadly. We can call parenthesis expansion the transition from an expression that contains negative numbers in parentheses to an expression that does not have parentheses. For example, we can go from 5 + (- 3) - (- 7) to 5 - 3 + 7. In fact, this is also a parenthesis expansion.

In the same way, we can replace the product of expressions in parentheses of the form (a + b) (c + d) with the sum a c + a d + b c + b d. This technique also does not contradict the meaning of parenthesis expansion.

Here's another example. We can assume that any expressions can be used instead of numbers and variables in expressions. For example, the expression x 2 1 a - x + sin (b) will correspond to an expression without parentheses of the form x 2 1 a - x 2 x + x 2 sin (b).

One more point deserves special attention, which concerns the peculiarities of recording decisions when opening parentheses. We can write the initial expression with parentheses and the result obtained after expanding the parentheses as equality. For example, after expanding the parentheses, instead of the expression 3 − (5 − 7) we get the expression 3 − 5 + 7 . We can write both of these expressions as the equality 3 - (5 - 7) = 3 - 5 + 7.

Carrying out actions with cumbersome expressions may require recording intermediate results. Then the solution will have the form of a chain of equalities. For instance, 5 − (3 − (2 − 1)) = 5 − (3 − 2 + 1) = 5 − 3 + 2 − 1 or 5 − (3 − (2 − 1)) = 5 − 3 + (2 − 1) = 5 − 3 + 2 − 1 .

Bracket expansion rules, examples

Let's start looking at the rules for expanding parentheses.

Single numbers in brackets

Negative numbers in parentheses are common in expressions. For example, (- 4) and 3 + (- 4). Positive numbers in brackets also have a place to be.

Let us formulate a rule for expanding brackets in which single positive numbers are enclosed. Suppose a is any positive number. Then (a) we can replace by a, + (a) by + a, - (a) by - a. If instead of a we take a specific number, then according to the rule: the number (5) will be written as 5 , the expression 3 + (5) without brackets takes the form 3 + 5 since + (5) is replaced by + 5 , and the expression 3 + (- 5) is equivalent to the expression 3 − 5 , because + (− 5) is replaced by − 5 .

Positive numbers are usually written without parentheses, since parentheses are unnecessary in this case.

Now consider the rule for expanding parentheses that contain a single negative number. + (- a) we replace with - a, - (- a) is replaced by + a. If the expression starts with a negative number (- a), which is written in parentheses, then the parentheses are omitted and instead of (- a) remains - a.

Here are some examples: (- 5) can be written as - 5, (- 3) + 0, 5 takes the form - 3 + 0, 5, 4 + (- 3) turns into 4 − 3 , and - (- 4) - (- 3) after expanding the brackets takes the form 4 + 3, since - (- 4) and - (- 3) is replaced with + 4 and + 3.

It should be understood that you cannot write the expression 3 · (- 5) as 3 · - 5. This will be discussed in the following paragraphs.

Let's see what the parenthesis expansion rules are based on.

According to the rule, the difference a - b is equal to a + (- b). Based on the properties of actions with numbers, we can form a chain of equalities (a + (- b)) + b = a + ((- b) + b) = a + 0 = a which will be fair. This chain of equalities, by virtue of the meaning of subtraction, proves that the expression a + (- b) is the difference a - b.

Based on the properties of opposite numbers and the rules for subtracting negative numbers, we can assert that - (- a) = a, a - (- b) = a + b.

There are expressions that are composed of a number, minus signs and several pairs of parentheses. Using the rules above allows you to consistently get rid of the brackets, moving from the inner brackets to the outer ones, or in the opposite direction. An example of such an expression would be - (- ((- (5)))). Let's open the brackets, moving from the inside to the outside: - (- ((- (5)))) = - (- ((- 5))) = - (- (- 5)) = - (5) = - 5. Also, this example can be parsed in the opposite direction: − (− ((− (5)))) = ((− (5))) = (− (5)) = − (5) = − 5 .

Under a and b, not only numbers can be understood, but also arbitrary numeric or literal expressions with a "+" sign in front, which are not sums or differences. In all these cases, you can apply the rules in the same way as we did for single numbers in brackets.

For example, after expanding the parentheses, the expression - (- 2 x) - (x 2) + (- 1 x) - (2 x y 2: z) takes the form 2 x - x 2 - 1 x - 2 x y 2: z. How did we do it? We know that - (- 2 x) is + 2 x, and since this expression is at the beginning, + 2 x can be written as 2 x, - (x 2) = - x 2, + (- 1 x) = - 1 x and - (2 x y 2: z) = - 2 x y 2: z.

In products of two numbers

Let's start with the rule for expanding parentheses in the product of two numbers.

Let's pretend that a and b are two positive numbers. In this case, the product of two negative numbers - a and - b of the form (- a) (- b) we can replace with (a b), and the products of two numbers with opposite signs of the form (- a) b and a (- a b)... Multiplying a minus by a minus gives a plus, and multiplying a minus by a plus, as well as multiplying a plus by a minus, gives a minus.

The correctness of the first part of the written rule is confirmed by the rule for multiplying negative numbers. To confirm the second part of the rule, we can use the rules for multiplying numbers with different signs.

Let's look at a few examples.

Example 1

Consider an algorithm for expanding parentheses in the product of two negative numbers - 4 3 5 and - 2, of the form (- 2) · - 4 3 5. To do this, replace the original expression with 2 · 4 3 5. Expand the parentheses and get 2 4 3 5.

And if we take the quotient of negative numbers (- 4): (- 2), then the record after expanding the brackets will look like 4: 2

In place of negative numbers - a and - b can be any expressions with a leading minus sign that are not sums or differences. For example, it can be products, quotients, fractions, powers, roots, logarithms, trigonometric functions, etc.

Expand the parentheses in the expression - 3 x x 2 + 1 x (- ln 5). According to the rule, we can perform the following transformations: - 3 x x 2 + 1 x (- ln 5) = - 3 x x 2 + 1 x ln 5 = 3 x x 2 + 1 x ln 5.

Expression (- 3) 2 can be converted to the expression (- 3 · 2). Then you can expand the brackets: - 3 2.

2 3 - 4 5 = - 2 3 4 5 = - 2 3 4 5

Division of numbers with different signs may also require parentheses to be expanded beforehand: (− 5) : 2 = (− 5: 2) = − 5: 2 and 2 3 4: (- 3, 5) = - 2 3 4: 3, 5 = - 2 3 4: 3, 5.

The rule can be used to perform multiplication and division on expressions with different signs. Here are two examples.

1 x + 1: x - 3 = - 1 x + 1: x - 3 = - 1 x + 1: x - 3

sin (x) (- x 2) = (- sin (x) x 2) = - sin (x) x 2

In products of three or more numbers

Let's move on to the product and quotients, which contain more numbers. The following rule will apply here for expanding parentheses. For an even number of negative numbers, you can omit the parentheses by replacing the numbers with their opposite. After that, you need to enclose the resulting expression in new parentheses. For an odd number of negative numbers, omitting the parentheses, replace the numbers with their opposite ones. After that, the resulting expression must be enclosed in new brackets and preceded by a minus sign.

Example 2

For example, let's take the expression 5 · (- 3) · (- 2), which is the product of three numbers. There are two negative numbers, therefore, we can write the expression as (5 · 3 · 2) and then finally open the parentheses, getting the expression 5 · 3 · 2.

In the product (- 2, 5) · (- 3): (- 2) · 4: (- 1, 25): (- 1) five numbers are negative. therefore (- 2, 5) (- 3): (- 2) 4: (- 1, 25): (- 1) = (- 2, 5 3: 2 4: 1, 25: 1) ... Finally, expanding the brackets, we obtain −2.5 3: 2 4: 1.25: 1.

The above rule can be substantiated as follows. First, we can rewrite such expressions as a product, replacing division by multiplication by the reciprocal. We represent each negative number as a product of a multiplier and replace - 1 or - 1 with (- 1) a.

Using the displacement property of multiplication, we swap the factors and transfer all factors equal to − 1 , to the beginning of the expression. The product of an even number minus ones is 1, and an odd number is equal to − 1 , which allows us to use the minus sign.

If we did not use the rule, then the chain of actions for expanding the parentheses in the expression - 2 3: (- 2) 4: - 6 7 would look like this:

2 3: (- 2) 4: - 6 7 = - 2 3 - 1 2 4 - 7 6 = = (- 1) 2 3 (- 1) 1 2 4 (- 1 ) 7 6 = = (- 1) (- 1) (- 1) 2 3 1 2 4 7 6 = (- 1) 2 3 1 2 4 7 6 = = - 2 3 1 2 4 7 6

The above rule can be used when expanding parentheses in expressions that are products and quotients with a minus sign that are not sums or differences. Take the expression

x 2 (- x): (- 1 x) x - 3: 2.

It can be reduced to an expression without parentheses x 2 x: 1 x x - 3: 2.

Expanding parentheses preceded by a + sign

Consider a rule that can be applied to expand parentheses that are preceded by a plus sign, and the "contents" of those parentheses are not multiplied or divisible by any number or expression.

According to the rule, the parentheses together with the character in front of them are omitted, while the signs of all terms in the parentheses are preserved. If there is no sign in front of the first term in parentheses, then you need to put a plus sign.

Example 3

For example, let's give the expression (12 − 3 , 5) − 7 ... Having omitted the parentheses, we keep the signs of the terms in parentheses and put a plus sign in front of the first term. The record will look like (12 - 3, 5) - 7 = + 12 - 3, 5 - 7. In the given example, it is not necessary to put a sign in front of the first term, since + 12 - 3, 5 - 7 = 12 - 3, 5 - 7.

Example 4

Let's take another example. Take the expression x + 2 a - 3 x 2 + 1 - x 2 - 4 + 1 x and perform actions with it x + 2 a - 3 x 2 + 1 - x 2 - 4 + 1 x = = x + 2 a - 3 x 2 + 1 - x 2 - 4 + 1 x

Here's another example of expanding parentheses:

Example 5

2 + x 2 + 1 x - x y z + 2 x - 1 + (- 1 + x - x 2) = = 2 + x 2 + 1 x - x y z + 2 x - 1 - 1 + x + x 2

How parentheses are expanded, preceded by a minus sign

Consider the cases where the parentheses are preceded by a minus sign, and which are not multiplied (or divided) by any number or expression. According to the rule of opening brackets preceded by a "-" sign, brackets with a "-" sign are omitted, while the signs of all terms inside the brackets are reversed.

Example 6

For example:

1 2 = 1 2, - 1 x + 1 = - 1 x + 1, - (- x 2) = x 2

Variable expressions can be converted using the same rule:

X + x 3 - 3 - - 2 x 2 + 3 x 3 x + 1 x - 1 - x + 2,

we get x - x 3 - 3 + 2 x 2 - 3 x 3 x + 1 x - 1 - x + 2.

Expansion of parentheses when multiplying a number by a parenthesis, expressions by a parenthesis

Here we will look at cases when you need to expand parentheses that are multiplied or divisible by some number or expression. Here formulas of the form (a 1 ± a 2 ±… ± a n) · b = (a 1 · b ± a 2 · b ±… ± a n · b) or b (a 1 ± a 2 ±… ± a n) = (b a 1 ± b a 2 ±… ± b a n), where a 1, a 2,…, a n and b are some numbers or expressions.

Example 7

For example, let's expand the parentheses in the expression (3 - 7) 2... According to the rule, we can carry out the following transformations: (3 - 7) 2 = (3 2 - 7 2). We get 3 2 - 7 2.

Expanding the parentheses in the expression 3 x 2 1 - x + 1 x + 2, we get 3 x 2 1 - 3 x 2 x + 3 x 2 1 x + 2.

Multiplying a parenthesis by a parenthesis

Consider the product of two brackets of the form (a 1 + a 2) · (b 1 + b 2). This will help us get a rule for expanding parentheses when performing parenthesis-to-parenthesis multiplication.

In order to solve the above example, we denote the expression (b 1 + b 2) like b. This will allow us to use the rule for multiplying the parenthesis by the expression. We get (a 1 + a 2) (b 1 + b 2) = (a 1 + a 2) b = (a 1 b + a 2 b) = a 1 b + a 2 b. Reverse replacement b by (b 1 + b 2), again apply the rule of multiplying the expression by the parenthesis: a 1 b + a 2 b = = a 1 (b 1 + b 2) + a 2 (b 1 + b 2) = = (a 1 b 1 + a 1 b 2) + (a 2 b 1 + a 2 b 2) = = a 1 b 1 + a 1 b 2 + a 2 b 1 + a 2 B 2

Thanks to a number of simple tricks, we can arrive at the sum of the products of each of the terms from the first parenthesis by each of the terms from the second parenthesis. The rule can be extended to any number of terms within parentheses.

Let us formulate the rules for multiplying a parenthesis by a parenthesis: in order to multiply two sums among themselves, it is necessary to multiply each of the terms of the first sum by each of the terms of the second sum and add the results obtained.

The formula will look like:

(a 1 + a 2 +... + a m) · (b 1 + b 2 +... + b n) = = a 1 b 1 + a 1 b 2 +. ... ... + a 1 b n + + a 2 b 1 + a 2 b 2 +. ... ... + a 2 b n + +. ... ... + + a m b 1 + a m b 1 +. ... ... a m b n

Let's expand the parentheses in the expression (1 + x) · (x 2 + x + 6) It is the product of two sums. Let's write the solution: (1 + x) (x 2 + x + 6) = = (1 x 2 + 1 x + 1 6 + x x 2 + x x + x 6) = = 1 x 2 + 1 x + 1 6 + x x 2 + x x + x 6

Separately, it is worth dwelling on those cases when the minus sign is present in the brackets along with the plus signs. For example, let's take the expression (1 - x) · (3 · x · y - 2 · x · y 3).

First, we represent the expressions in parentheses as sums: (1 + (- x)) (3 x y + (- 2 x y 3))... Now we can apply the rule: (1 + (- x)) (3 x y + (- 2 x y 3)) = (1 3 x y + 1 (- 2 x Y 3) + (- x) 3 x y + (- x) (- 2 x y 3))

Expand the parentheses: 1 3 x y - 1 2 x y 3 - x 3 x y + x 2 x y 3.

Expansion of brackets in products of multiple brackets and expressions

If there are three or more expressions in parentheses in an expression, the parentheses must be expanded sequentially. It is necessary to start the transformation by putting the first two factors in parentheses. Within these brackets, we can perform transformations according to the rules discussed above. For example, parentheses in the expression (2 + 4) · 3 · (5 + 7 · 8).

The expression contains three factors at once (2 + 4) , 3 and (5 + 7 8). We will expand the parentheses sequentially. Let's enclose the first two factors in one more parentheses, which we will make red for clarity: (2 + 4) 3 (5 + 7 8) = ((2 + 4) 3) (5 + 7 8).

In accordance with the rule for multiplying a parenthesis by a number, we can carry out the following actions: ((2 + 4) 3) (5 + 7 8) = (2 3 + 4 3) (5 + 7 8).

Multiply the parenthesis by the parenthesis: (2 3 + 4 3) (5 + 7 8) = 2 3 5 + 2 3 7 8 + 4 3 5 + 4 3 7 8 ...

Bracket in natural degree

Degrees, the bases of which are some expressions written in brackets, with natural indicators can be considered as the product of several brackets. Moreover, according to the rules from the two previous paragraphs, they can be written without these brackets.

Consider the process of converting an expression (a + b + c) 2. It can be written as a product of two brackets (a + b + c) (a + b + c)... Let's multiply the parenthesis by the parenthesis and get a · a + a · b + a · c + b · a + b · b + b · c + c · a + c · b + c · c.

Let's take another example:

Example 8

1 x + 2 3 = 1 x + 2 1 x + 2 1 x + 2 = = 1 x 1 x + 1 x 2 + 2 1 x + 2 2 1 x + 2 = = 1 x 1 x 1 x + 1 x 2 1 x + 2 1 x 1 x + 2 2 1 x + 1 x 1 x 2 + + 1 x 2 2 + 2 1 x 2 + 2 2 2

Divide a parenthesis by a number and parentheses by a parenthesis

Dividing a parenthesis by a number assumes that you must divide all the terms in parentheses by a number. For example, (x 2 - x): 4 = x 2: 4 - x: 4.

Division can be previously replaced by multiplication, after which you can use the appropriate rule for expanding parentheses in the product. The same rule applies when dividing a parenthesis by a parenthesis.

For example, we need to expand the parentheses in the expression (x + 2): 2 3. To do this, first replace the division by multiplication by the inverse number (x + 2): 2 3 = (x + 2) · 2 3. Multiply the parenthesis by the number (x + 2) 2 3 = x 2 3 + 2 2 3.

Here's another example of division by parenthesis:

Example 9

1 x + x + 1: (x + 2).

Replace division by multiplication: 1 x + x + 1 · 1 x + 2.

Perform the multiplication: 1 x + x + 1 1 x + 2 = 1 x 1 x + 2 + x 1 x + 2 + 1 1 x + 2.

Bracket Expansion Order

Now let's consider the order of applying the rules discussed above in general expressions, i.e. in expressions that contain sums with differences, products with quotients, parentheses in natural degree.

Procedure for performing actions:

• the first step is to raise the parentheses to the natural degree;
• at the second stage, brackets are opened in works and private ones;
• the final step is to expand the parentheses in the sums and differences.

Let us consider the procedure for performing actions using the example of the expression (- 5) + 3 · (- 2): (- 4) - 6 · (- 7). Let's transform from expressions 3 (- 2): (- 4) and 6 (- 7), which should take the form (3 2: 4) and (- 6 7). Substituting the results obtained into the original expression, we get: (- 5) + 3 (- 2): (- 4) - 6 (- 7) = (- 5) + (3 2: 4) - (- 6 7). We open the brackets: - 5 + 3 2: 4 + 6 7.

When dealing with expressions that contain parentheses in parentheses, it is convenient to transform from the inside out.

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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 designations, brackets, roots, special mathematical signs, degrees, functions, etc. The whole theory, according to tradition, will be supplied with abundant and detailed examples.

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How do I find the value of a numeric expression?

Numeric expressions, among other things, help describe a problem condition 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, powers, roots, brackets, etc. Within the framework of a task, it is often necessary to find the meaning of an expression. How to do this will be discussed below.

The simplest cases

These are the cases when 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 brackets, as well as the ability to perform operations with different 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. Here are some 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 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 the 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 value you were looking for was found.

Expressions with brackets

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

0.5 (0.76 - 0.06) = 0.50.7 = 0.35.

The meaning of expressions containing parentheses in parentheses follows 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 the actions starting with the innermost brackets, moving on 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.

Rooted expressions

The mathematical expressions we need to find the values ​​for can contain root signs. Moreover, the expression itself can be under the root sign. What should be done in this case? First, you need to find the value of the expression under the root, and then extract the root from the resulting number. Whenever 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 you can evaluate 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 a rooted expression often requires converting the original expression first. Let us explain this with one more example.

Example 6. The value of a numeric expression

How much is 3 + 1 3 - 1 - 1

As you can see, there is no way for us to replace the root with an exact value, which complicates the calculation process. However, in in this case you can apply the abbreviated multiplication formula.

3 + 1 3 - 1 = 3 - 1 .

In this way:

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

Power expressions

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

Example 7. 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 degree properties.

Example 8. 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 meaning.

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

Fraction expressions

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 a fraction, then the values ​​of these expressions are first calculated, and the final value of the fraction itself is written. Arithmetic operations are performed in a standard manner. Let's consider the solution of an example.

Example 9. Value of a numeric expression

Find the value of the expression containing the 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 calculate their values ​​first.

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, it is best to simplify any expression to the maximum, reducing all calculations to the simplest cases.

Example 10. Value of a numeric expression

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

We cannot extract the root of five entirely, but we can simplify the original expression by transforming it.

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 the 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.

Numerical 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. 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 not possible to calculate the exact value of the logarithm, simplifying the expression helps you find its value.

Example 11. 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.

By 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 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 an expression contains trigonometric functions of sine, cosine, tangent and cotangent, as well as functions that are inverse to them. The values ​​are computed from before all other arithmetic operations are performed. Otherwise, the expression is simplified.

Example 12. 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.

Expression value found.

Often, in order to find the value of an expression with trigonometric functions, it must first be transformed. Let us explain with an example.

Example 13. Value of a numeric expression

You 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 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.

The general case of a numeric expression

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

How to find the meaning of an expression

1. Roots, degrees, logarithms, etc. are replaced by their values.
2. Actions in parentheses are performed.
3. 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. Value of a numeric expression

Let us 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 rather complex and cumbersome. It was not by chance that we chose just such an example, trying to fit all the cases described above into it. How do you find the meaning 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 consistently 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 the fraction, everything is simpler:

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, let's 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 the exact values ​​of roots, logarithms, sines, etc. If this is not possible, you can try to get rid of them by mathematical transformations.

Calculating the values ​​of expressions in rational ways

Calculate numeric values ​​consistently and accurately. This process can be rationalized and accelerated by using various properties of actions 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. 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. In this case, 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 property of subtracting equal numbers. Without performing any action, 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 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 parentheses. A rational approach to calculating expressions with fractions is to reduce the same 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 actions in parentheses, but reducing the fraction, we can say that the value of the expression is 1 3.

Finding the values ​​of expressions with variables

The meaning of an alphabetic expression and an expression with variables is found for specific specified 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 specified values ​​of letters and variables into the original expression, and then calculate the value of the resulting numerical expression.

Example 15. Value of an expression with variables

Evaluate the value of 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 in such a way as to 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 you don't need 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.

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

So, if a numerical expression is composed of numbers and signs +, -, · and:, then in order from left to right, you must first perform multiplication and division, and then addition and subtraction, which will allow you to find the desired value of the expression.

Let's give a solution of examples for clarification.

Example.

Evaluate the value of the expression 14−2 · 15: 6−3.

Solution.

To find the value of an expression, you need to perform all the actions indicated in it in accordance with the accepted order of performing these actions. First, in order from left to right, we perform multiplication and division, we get 14-215: 6-3 = 14-30: 6-3 = 14-5-3... Now, also, in order from left to right, we perform the remaining actions: 14−5−3 = 9−3 = 6. So we found the value of the original expression, it is 6.

Answer:

14-215: 6-3 = 6.

Example.

Find the meaning of the expression.

Solution.

In this example, we first need to perform the multiplication 2 · (−7) and division and multiplication in the expression. Remembering how it is done, we find 2 (−7) = - 14. And to perform actions in the expression, first , then , and execute: .

Substitute the obtained values ​​into the original expression:.

But what if there is a numerical expression under the root sign? To get the value of such a root, you must first find the value of the radical expression, adhering to the accepted order of execution of actions. For instance, .

In numerical expressions, the roots should be perceived as some numbers, and it is advisable to immediately replace the roots with their values, and then find the value of the resulting expression without roots, performing actions in the accepted sequence.

Example.

Find the meaning of the expression with roots.

Solution.

First, we find the value of the root ... To do this, first, we calculate the value of the radical expression, we have −2 3−1 + 60: 4 = −6−1 + 15 = 8... And secondly, we find the value of the root.

Now let's calculate the value of the second root from the original expression:.

Finally, we can find the value of the original expression by replacing the roots with their values:.

Answer:

Quite often, to make it possible to find the value of an expression with roots, you first have to transform it. Let's show the solution of an example.

Example.

What is the meaning of the expression .

Solution.

We cannot replace the root of three with its exact value, which does not allow us to calculate the value of this expression in the way described above. However, we can compute the value of this expression by performing simple transformations. Applicable difference of squares formula:. Considering, we get ... Thus, the value of the original expression is 1.

Answer:

.

With degrees

If the base and the exponent are numbers, then their value is calculated according to the definition of the exponent, for example, 3 2 = 3 · 3 = 9 or 8 −1 = 1/8. There are also records when the base and / or exponent are some expressions. In these cases, you need to find the value of the expression in the base, the value of the expression in the exponent, and then calculate the value of the degree itself.

Example.

Find the value of an expression with powers of the form 2 3 4-10 + 16 (1-1 / 2) 3.5-2 1/4.

Solution.

In the original expression, two degrees are 2 3 4-10 and (1-1 / 2) 3.5-2 1/4. Their values ​​must be calculated before performing any other steps.

Let's start with a power of 2 3 4−10. In its indicator there is a numerical expression, we calculate its value: 3 4-10 = 12-10 = 2. Now you can find the value of the degree itself: 2 3 4−10 = 2 2 = 4.

At the base and the exponent (1-1 / 2) 3.5-2 We have (1-1 / 2) 3.5-21 / 4 = (1/2) 3 = 1/8.

Now we return to the original expression, replace the powers in it with their values, and find the value of the expression we need: 2 3 4−10 + 16 (1−1 / 2) 3.5−2 1/4 = 4 + 16 1/8 = 4 + 2 = 6.

Answer:

2 3 4−10 + 16 (1−1 / 2) 3.5−2 1/4 = 6.

It is worth noting that there are more common cases when it is advisable to conduct a preliminary simplification of expression with powers on the base .

Example.

Find the meaning of the expression .

Solution.

Judging by the exponents in this expression, the exact values ​​of the exponents cannot be obtained. Let's try to simplify the original expression, maybe this will help find its meaning. We have

Answer:

.

Degrees in expressions often go hand in hand with logarithms, but we will talk about finding the values ​​of expressions with logarithms in one of the.

Finding the value of an expression with fractions

Numeric expressions in their notation can contain fractions. When you need to find the meaning of such an expression, fractions other than ordinary fractions should be replaced with their values ​​before performing the rest of the steps.

The numerator and denominator of fractions (which are different from ordinary fractions) can contain both some numbers and expressions. To calculate the value of such a fraction, you need to calculate the value of the expression in the numerator, calculate the value of the expression in the denominator, and then calculate the value of the fraction itself. This order is explained by the fact that the fraction a / b, where a and b are some expressions, is essentially a quotient of the form (a) :( b), since.

Let's consider the solution of an example.

Example.

Find the meaning of an expression with fractions .

Solution.

In the original numerical expression, three fractions and . To find the value of the original expression, we first need these fractions, replace them with values. Let's do it.

The numerator and denominator of the fraction contains numbers. To find the value of such a fraction, replace the fractional bar with a division sign, and perform this action: .

The numerator of the fraction contains the expression 7−2 · 3, its value is easy to find: 7−2 · 3 = 7−6 = 1. In this way, . You can proceed to finding the value of the third fraction.

The third fraction in the numerator and denominator contains numerical expressions, therefore, first you need to calculate their values, and this will allow you to find the value of the fraction itself. We have .

It remains to substitute the found values ​​into the original expression, and perform the remaining actions:.

Answer:

.

Often, when finding the values ​​of expressions with fractions, you have to do simplification of fractional expressions based on performing actions with fractions and reducing fractions.

Example.

Find the meaning of the expression .

Solution.

The root of five is not entirely extracted, so to find the value of the original expression, let's first simplify it. For this get rid of irrationality in the denominator first fraction: ... After that, the original expression will take the form ... After subtracting the fractions, the roots will disappear, which will allow us to find the value of the initially specified expression:.

Answer:

.

With logarithms

If the numeric expression contains, and if it is possible to get rid of them, then this is done before performing the rest of the actions. For example, when you find the value of the expression log 2 4 + 2 + 6 = 8.

When there are numerical expressions under the sign of the logarithm and / or at its base, their values ​​are first found, after which the value of the logarithm is calculated. For example, consider an expression with a logarithm of the form ... At the base of the logarithm and under its sign there are numerical expressions, we find their values:. Now we find the logarithm, after which we complete the calculations:.

If the logarithms are not calculated exactly, then simplifying the initial expression using it can help to find the value of the original expression. At the same time, you need to have a good command of the article material. converting logarithmic expressions.

Example.

Find the value of an expression with logarithms .

Solution.

Let's start by calculating log 2 (log 2 256). Since 256 = 2 8, then log 2 256 = 8, therefore log 2 (log 2 256) = log 2 8 = log 2 2 3 = 3.

The logarithms of log 6 2 and log 6 3 can be grouped. The sum of the logarithms of log 6 2 + log 6 3 is equal to the logarithm of the product log 6 (2 3), so log 6 2 + log 6 3 = log 6 (2 3) = log 6 6 = 1.

Now let's deal with the fraction. To begin with, we rewrite the base of the logarithm in the denominator as an ordinary fraction as 1/5, after which we will use the properties of logarithms, which will allow us to get the value of the fraction:
.

It remains only to substitute the obtained results into the original expression and finish finding its value:

Answer:

How do I find the value of a trigonometric expression?

When a numeric expression contains or, etc., their values ​​are calculated before performing other actions. If there are numerical expressions under the sign of trigonometric functions, then their values ​​are first calculated, after which the values ​​of trigonometric functions are found.

Example.

Find the meaning of the expression .

Solution.

Referring to the article, we get and cosπ = −1. We substitute these values ​​into the original expression, it takes the form ... To find its value, you first need to perform exponentiation, and then finish the calculations:.

Answer:

.

It should be noted that the calculation of the values ​​of expressions with sines, cosines, etc. often requires prior converting trigonometric expression.

Example.

What is the value of a trigonometric expression .

Solution.

We transform the original expression using, in this case, we need the formula for the cosine of a double angle and the formula for the cosine of the sum:

The performed transformations helped us find the meaning of the expression.

Answer:

.

General case

In general, a numeric expression can contain roots, powers, fractions, functions, and brackets. Finding the values ​​of such expressions is to do the following:

• first roots, powers, fractions, etc. are replaced by their values,
• further actions in brackets,
• and in order from left to right, the remaining operations are performed - multiplication and division, followed by addition and subtraction.

The listed actions are performed until the final result is obtained.

Example.

Find the meaning of the expression .

Solution.

The form of this expression is rather complicated. In this expression, we see fraction, roots, degrees, sine and logarithm. How do you find its meaning?

Moving along the record from left to right, we come across a fraction of the form ... We know that when working with complex fractions, we need to separately calculate the value of the numerator, separately - the denominator, and, finally, find the value of the fraction.

In the numerator we have a root of the form ... To determine its value, you first need to calculate the value of the radical expression ... There is a sine here. We can find its value only after calculating the value of the expression ... We can do this:. Then, whence and .

The denominator is simple:.

In this way, .

After substituting this result into the original expression, it will take the form. The resulting expression contains the degree. To find its value, you first have to find the value of the indicator, we have .

So, .

Answer:

.

If it is not possible to calculate the exact values ​​of the roots, degrees, etc., then you can try to get rid of them using some transformations, and then return to calculating the value according to the indicated scheme.

Rational ways of calculating the values ​​of expressions

Computing the values ​​of numeric expressions requires consistency and care. Yes, you must adhere to the sequence of actions written in the previous paragraphs, but you do not need to do it blindly and mechanically. By this we mean that it is often possible to rationalize the process of finding the meaning of an expression. For example, some properties of actions with numbers can significantly speed up and simplify finding the value of an expression.

For example, we know this property of multiplication: if one of the factors in the product is zero, then the value of the product is zero. Using this property, we can immediately say that the value of the expression 0 (2 3 + 893-3234: 54 65-79 56 2.2)(45 36−2 4 + 456: 3 43) is equal to zero. If we adhered to the standard order of performing actions, then first we would have to calculate the values ​​of bulky expressions in parentheses, and this would take a lot of time, and the result would still be zero.

It is also convenient to use the property of subtracting equal numbers: if you subtract an equal number from a number, the result will be zero. This property can be considered more broadly: the difference between two identical numerical expressions is zero. For example, without evaluating the values ​​of the expressions in parentheses, you can find the value of the expression (54 6−12 47362: 3) - (54 6−12 47362: 3), it is equal to zero, since the original expression is the difference of the same expressions.

Identical transformations can contribute to the rational calculation of the values ​​of expressions. For example, the grouping of terms and factors can be useful, and brackets are also often used. So the value of the expression 53 5 + 53 7−53 11 + 5 is very easy to find after putting the factor 53 outside the brackets: 53 (5 + 7−11) + 5 = 53 1 + 5 = 53 + 5 = 58... Calculating directly would take much longer.

In conclusion of this paragraph, let us pay attention to a rational approach to calculating the values ​​of expressions with fractions - the same factors in the numerator and denominator of a fraction are canceled. For example, canceling the same expressions in the numerator and denominator of a fraction allows you to immediately find its value, which is 1/2.

Finding the value of a literal expression and an expression with variables

The meaning of an alphabetic expression and an expression with variables is found for specific specified values ​​of letters and variables. That is, we are talking about finding the value of a literal expression for given values ​​of letters or about finding the value of an expression with variables for selected values ​​of variables.

The rule Finding the value of a literal expression or an expression with variables for given values ​​of letters or selected values ​​of variables is as follows: you need to substitute these values ​​of letters or variables into the original expression, and calculate the value of the resulting numerical expression, it is the desired value.

Example.

Evaluate the expression 0.5 x − y at x = 2.4 and y = 5.

Solution.

To find the required value of the expression, you first need to substitute these values ​​of the variables into the original expression, and then perform the following steps: 0.5 · 2.4-5 = 1.2-5 = −3.8.

Answer:

−3,8 .

In conclusion, we note that sometimes performing transformations of literal expressions and expressions with variables allows you to get their values, regardless of the values ​​of letters and variables. For example, the expression x + 3 − x can be simplified, after which it becomes 3. Hence, we can conclude that the value of the expression x + 3 − x is equal to 3 for any values ​​of the variable x from its range of permissible values ​​(ODV). Another example: the value of the expression is equal to 1 for all positive values ​​of x, so the range of valid values ​​of the variable x in the original expression is the set of positive numbers, and equality takes place in this range.

Bibliography.

• Mathematics: textbook. for 5 cl. general education. institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., Erased. - M .: Mnemosina, 2007 .-- 280 p .: ill. ISBN 5-346-00699-0.
• Mathematics. Grade 6: textbook. for general education. institutions / [N. Ya. Vilenkin and others]. - 22nd ed., Rev. - M .: Mnemosina, 2008 .-- 288 p.: Ill. ISBN 978-5-346-00897-2.
• Algebra: study. for 7 cl. 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: study. for 8 cl. 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.
• Algebra: Grade 9: textbook. for general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M.: Education, 2009 .-- 271 p. : ill. - ISBN 978-5-09-021134-5.
• Algebra and the beginning of the analysis: Textbook. for 10-11 cl. general education. institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M .: Education, 2004. - 384 p .: ill. - ISBN 5-09-013651-3.

Numeric expression Is any record of numbers, arithmetic signs and brackets. A numeric expression can consist of just one number. Let's remind that the main arithmetic operations are “addition”, “subtraction”, “multiplication” and “division”. These actions correspond to the signs "+", "-", "∙", ":".

Of course, in order for us to get a numerical expression, notation of numbers and arithmetic signs must be meaningful. So, for example, such a notation 5: + ∙ cannot be called a numerical expression, since this is a random set of characters that has no meaning. On the contrary, 5 + 8 ∙ 9 is already a real numerical expression.

The value of a numeric expression.

Let's say right away that if we perform the actions indicated in a numerical expression, then we will get a number as a result. This number is called the value of a numeric expression.

Let's try to calculate what we get as a result of performing the actions of our example. According to the order in which the arithmetic operations are performed, we first perform the multiplication operation. Multiply 8 by 9. Get 72. Now add 72 and 5. Get 77.
So 77 - meaning numerical expression 5 + 8 ∙ 9.

Numerical equality.

You can write it this way: 5 + 8 ∙ 9 = 77. Here we first used the sign "=" ("Equal"). Such a notation, in which two numerical expressions are separated by the "=" sign, is called numerical equality... Moreover, if the values ​​of the left and right sides of the equality coincide, then the equality is called faithful... 5 + 8 ∙ 9 = 77 - true equality.
If we write 5 + 8 ∙ 9 = 100, then it will already be false equality, since the values ​​of the left and right sides of this equality no longer coincide.

It should be noted that in a numeric expression, we can also use parentheses. The parentheses affect the order in which actions are performed. So, for example, let's modify our example by adding brackets: (5 + 8) ∙ 9. Now, first you need to add 5 and 8. We get 13. And then multiply 13 by 9. We get 117. Thus, (5 + 8) ∙ 9 = 117.
117 – meaning numerical expression (5 + 8) ∙ 9.

To read an expression correctly, you need to determine which action is performed last to calculate the value of a given numeric expression. So, if the last action is subtraction, then the expression is called "difference". Accordingly, if the last action is sum - "sum", division - "quotient", multiplication - "product", exponentiation - "degree".

For example, the numerical expression (1 + 5) (10-3) reads like this: "the product of the sum of the numbers 1 and 5 by the difference between the numbers 10 and 3".

Examples of numeric expressions.

Here's an example of a more complex numeric expression:

\ [\ left (\ frac (1) (4) +3.75 \ right): \ frac (1.25 + 3.47 + 4.75-1.47) (4 \ centerdot 0.5) \]

This numeric expression uses prime numbers, fractions, and decimals. The signs of addition, subtraction, multiplication and division are also used. The fraction bar also replaces the division sign. Despite the seeming complexity, it is quite easy to find the value of this numerical expression. The main thing is to be able to perform operations with fractions, as well as to carefully and accurately do calculations, observing the order of performing actions.

In parentheses, we have the expression $ \ frac (1) (4) + 3.75 $. Convert the decimal 3.75 to a fraction.

$ 3.75 = 3 \ frac (75) (100) = 3 \ frac (3) (4) $

So, $ \ frac (1) (4) + 3.75 = \ frac (1) (4) +3 \ frac (3) (4) = 4 $

Further, in the numerator of the fraction \ [\ frac (1.25 + 3.47 + 4.75-1.47) (4 \ centerdot 0.5) \] we have the expression 1.25 + 3.47 + 4.75-1.47. To simplify this expression, we apply the displacement law of addition, which says: "The sum does not change from the change of places of the terms." That is, 1.25 + 3.47 + 4.75-1.47 = 1.25 + 4.75 + 3.47-1.47 = 6 + 2 = 8.

In the denominator of the fraction, the expression $ 4 \ centerdot 0.5 = 4 \ centerdot \ frac (1) (2) = 4: 2 = 2 $

We get $ \ left (\ frac (1) (4) +3.75 \ right): \ frac (1.25 + 3.47 + 4.75-1.47) (4 \ centerdot 0.5) = 4: \ frac (8) (2) = 4: 4 = 1 $

When are numeric expressions meaningless?

Let's take another example. In the denominator of the fraction $ \ frac (5 + 5) (3 \ centerdot 3-9) $ the value of the expression $ 3 \ centerdot 3-9 $ is 0. And, as we know, division by zero is impossible. Therefore, the fraction $ \ frac (5 + 5) (3 \ centerdot 3-9) $ has no value. Numeric expressions that have no meaning are said to be "meaningless."

If in numerical expression, in addition to numbers, we use letters, then we will get already