EntranceLet's find the equal value of the expression. Finding the meaning of an expression: rules, examples, solutions
So, if a numerical expression is made up of numbers and the 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 some examples for clarification.
Example.
Calculate 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 specified 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 14−2·15:6−3=14−30:6−3=14−5−3. Now we also perform the remaining actions in order from left to right: 14−5−3=9−3=6. This is how we found the value of the original expression, it is equal to 6.
Answer:
14−2·15:6−3=6.
Example.
Find the meaning of the expression.
Solution.
In this example, we first need to do the multiplication 2·(−7) and the division with the multiplication in the expression . Remembering how , we find 2·(−7)=−14. And to perform the actions in the expression first 
, then 
, and execute: 
.
We substitute the obtained values into the original expression: .
But what if there is a numerical expression under the root sign? To obtain the value of such a root, you must first find the value of the radical expression, adhering to the accepted order of performing actions. For example, .
In numerical expressions, 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 let's find the value of the root 
. To do this, firstly, 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 meaning of the original expression by replacing the roots with their values: .
Answer:
Quite often, in order to find the meaning of an expression with roots, it is first necessary to transform it. Let's show the solution of the example.
Example.
What is the meaning of the expression 
.
Solution.
We are not able to replace the root of three with its exact value, which does not allow us to calculate the value of this expression in the manner described above. However, we can calculate the value of this expression by performing simple transformations. Applicable square difference formula: . Taking into account , we get 
. Thus, the value of the original expression is 1.
Answer:
.
With degrees
If the base and exponent are numbers, then their value is calculated by determining the degree, for example, 3 2 =3·3=9 or 8 −1 =1/8. There are also entries where 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 there are two powers 2 3·4−10 and (1−1/2) 3.5−2·1/4. Their values must be calculated before performing other actions.
Let's start with the power 2 3·4−10. Its indicator contains a numerical expression, let's 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.
The base and exponent (1−1/2) 3.5−2 1/4 contain expressions; we calculate their values in order to then find the value of the exponent. We have (1−1/2) 3.5−2 1/4 =(1/2) 3 =1/8.
Now we return to the original expression, replace the degrees 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, it will not be possible to obtain exact values of the exponents. Let's try to simplify the original expression, maybe this will help find its meaning. We have 
Answer:
.
Powers in expressions often go hand in hand with logarithms, but we will talk about finding the meaning of expressions with logarithms in one of the.
Finding the value of an expression with fractions
Numeric expressions may contain fractions in their notation. When you need to find the value of such an expression, fractions other than ordinary fractions, you should replace them 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, essentially represents a quotient of the form (a):(b), since .
Let's look at the example solution.
Example.
Find the meaning of an expression with fractions 
.
Solution.
There are three fractions in the original numerical expression 
And . To find the value of the original expression, we first need to replace these fractions with their values. Let's do it.
The numerator and denominator of a fraction contain numbers. To find the value of such a fraction, replace the fraction bar with a division sign and perform this action: 
.
In the numerator of the fraction there is an expression 7−2·3, its value is easy to find: 7−2·3=7−6=1. Thus, . You can proceed to finding the value of the third fraction.
The third fraction in the numerator and denominator contains numerical expressions, therefore, you first 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 perform simplifying fractional expressions, based on performing operations with fractions and reducing fractions.
Example.
Find the meaning of the expression 
.
Solution.
The root of five cannot be extracted completely, so to find the value of the original expression, let’s first simplify it. For this let's get rid of irrationality in the denominator first fraction: 
. After this, 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 given expression: .
Answer:
.
With logarithms
If a numeric expression contains , and if it is possible to get rid of them, then this is done before performing other actions. For example, when finding the value of the expression log 2 4+2·3, the logarithm log 2 4 is replaced by its value 2, after which the remaining actions are performed in the usual order, that is, log 2 4+2·3=2+2·3=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 logarithms are not calculated accurately, then preliminary simplification of it using . In this case, 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 log 6 2 and log 6 3 can be grouped. The sum of the logarithms log 6 2+log 6 3 is equal to the logarithm of the product log 6 (2 3), thus, log 6 2+log 6 3=log 6 (2 3)=log 6 6=1.
Now let's look at the fraction. To begin with, we will rewrite the base of the logarithm in the denominator in the form of an ordinary fraction as 1/5, after which we will use the properties of logarithms, which will allow us to obtain the value of the fraction: 
.
All that remains is to substitute the results obtained into the original expression and finish finding its value: 
Answer:
How to 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 the trigonometric functions are found.
Example.
Find the meaning of the expression 
.
Solution.
Turning 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 is worth noting that calculating the values of expressions with sines, cosines, etc. often requires prior converting a trigonometric expression.
Example.
What is the value of the trigonometric expression 
.
Solution.
Let's transform the original expression using , in this case we will need the double angle cosine formula and the sum cosine formula: 
The transformations we made helped us find the meaning of the expression.
Answer:
.
General case
In general, a numerical expression can contain roots, powers, fractions, some functions, and parentheses. Finding the values of such expressions consists of performing the following actions:
- 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 quite complex. In this expression we see fractions, roots, powers, sine and logarithms. How to find its value?
Moving through the record from left to right, we come across a fraction of the form 
. We know that when working with fractions complex type, 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 the 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 
. This we can do: . Then where and from 
.
The denominator is simple: .
Thus, 
.
After substituting this result into the original expression, it will take the form . The resulting expression contains the degree . To find its value, we first have to find the value of the indicator, we have 
.
So, .
Answer:
.
If it is not possible to calculate the exact values of roots, powers, etc., then you can try to get rid of them using some transformations, and then return to calculating the value according to the specified scheme.
Rational ways to calculate the values of expressions
Calculating the values of numeric expressions requires consistency and accuracy. Yes, it is necessary to adhere to the sequence of actions recorded in the previous paragraphs, but there is no need to do this blindly and mechanically. What we mean by this is that it is often possible to rationalize the process of finding the meaning of an expression. For example, certain properties of operations 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 equal to zero, then the value of the product is equal to 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 followed the standard order of operations, we would first have to calculate the values of the cumbersome expressions in parentheses, which would take a lot of time, and the result would still be zero.
It is also convenient to use the subtraction property equal numbers: If you subtract an equal number from a number, the result is zero. This property can be considered more broadly: the difference between two identical numerical expressions is zero. For example, without calculating the value 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 identical expressions.
Identity transformations can facilitate the rational calculation of expression values. For example, grouping terms and factors can be useful; putting the common factor out of brackets is no less often used. So the value of the expression 53·5+53·7−53·11+5 is very easy to find after taking the factor 53 out of brackets: 53·(5+7−11)+5=53·1+5=53+5=58. Direct calculation would have taken much longer.
To conclude this point, let us pay attention to a rational approach to calculating the values of expressions with fractions - identical factors in the numerator and denominator of the fraction are canceled. For example, reducing the same expressions in the numerator and denominator of a fraction 
allows you to immediately find its value, which is equal to 1/2.
Finding the value of a literal expression and an expression with variables
The value of a literal expression and an expression with variables is found for specific given values of letters and variables. That is, we are talking about finding the value of a literal expression for given letter values, or about finding the value of an expression with variables for selected variable values.
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 the given values of letters or variables into the original expression, and calculate the value of the resulting numeric expression; it is the desired value.
Example.
Calculate the value of 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 the given 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 .
As a final note, sometimes performing conversions on literal and variable expressions will yield their values, regardless of the values of the letters and variables. For example, the expression x+3−x can be simplified, after which it will take the form 3. From this 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 (APV). Another example: the value of the expression is equal to 1 for all positive values of x, so the range of permissible values of the variable x in the original expression is the set of positive numbers, and in this range the equality holds.
Bibliography.
- Mathematics: textbook for 5th grade. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 pp.: ill. ISBN 5-346-00699-0.
- Mathematics. 6th grade: educational. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
- Algebra: textbook for 7th grade general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 17th ed. - M.: Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
- Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
- Algebra: 9th grade: educational. for general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2009. - 271 p. : ill. - ISBN 978-5-09-021134-5.
- Algebra and the beginning of analysis: Proc. for 10-11 grades. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M.: Education, 2004. - 384 pp.: ill. - ISBN 5-09-013651-3.
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 give an expression containing letter designations, brackets, roots, special mathematical symbols, powers, functions, etc. As per tradition, we will provide the entire theory with abundant and detailed examples.
How to find the value of a numeric expression?
Numerical expressions, among other things, help to describe the condition of a problem in mathematical language. In general, mathematical expressions can be either very simple, consisting of a pair of numbers and arithmetic symbols, or very complex, containing functions, powers, roots, parentheses, etc. As part of a task, it is often necessary to find the meaning of a particular expression. How to do this will be discussed below.
The simplest cases
These are cases where the expression contains nothing but numbers and arithmetic operations. To successfully find the values of such expressions, you will need knowledge of the order of performing arithmetic operations without parentheses, as well as the ability to perform operations with various numbers.
If the expression contains only numbers and arithmetic signs " + " , " · " , " - " , " ÷ " , then the actions are performed from left to right in the following order: first multiplication and division, then addition and subtraction. Let's give examples.
Example 1: The value of a numeric expression
Let you need to find the values of the expression 14 - 2 · 15 ÷ 6 - 3.
Let's do the multiplication and division first. We get:
14 - 2 15 ÷ 6 - 3 = 14 - 30 ÷ 6 - 3 = 14 - 5 - 3.
Now we carry out the subtraction and get the final result:
14 - 5 - 3 = 9 - 3 = 6 .
Example 2: The value of a numeric expression
Let's calculate: 0, 5 - 2 · - 7 + 2 3 ÷ 2 3 4 · 11 12.
First we perform fraction conversion, division and multiplication:
0, 5 - 2 · - 7 + 2 3 ÷ 2 3 4 · 11 12 = 1 2 - (- 14) + 2 3 ÷ 11 4 · 11 12
1 2 - (- 14) + 2 3 ÷ 11 4 11 12 = 1 2 - (- 14) + 2 3 4 11 11 12 = 1 2 - (- 14) + 2 9.
Now let's do some addition and subtraction. Let's group the fractions and bring them to a common denominator:
1 2 - (- 14) + 2 9 = 1 2 + 14 + 2 9 = 14 + 13 18 = 14 13 18 .
The required value has been found.
Expressions with parentheses
If an expression contains parentheses, they define the order of operations in that expression. The actions in brackets are performed first, and then all the others. Let's show this with an example.
Example 3: The value of a numeric expression
Let's find the value of the expression 0.5 · (0.76 - 0.06).
The expression contains parentheses, so we first perform the subtraction operation in parentheses, and only then the multiplication.
0.5 · (0.76 - 0.06) = 0.5 · 0.7 = 0.35.
The meaning of expressions containing parentheses within parentheses is found according to the same principle.
Example 4: The value of a numeric expression
Let's calculate the value 1 + 2 1 + 2 1 + 2 1 - 1 4.
We will perform actions starting from the innermost brackets, moving to the outer ones.
1 + 2 1 + 2 1 + 2 1 - 1 4 = 1 + 2 1 + 2 1 + 2 3 4
1 + 2 1 + 2 1 + 2 3 4 = 1 + 2 1 + 2 2, 5 = 1 + 2 6 = 13.
When finding the meanings of expressions with brackets, the main thing is to follow the sequence of actions.
Expressions with roots
Mathematical expressions whose values we need to find may contain root signs. Moreover, the expression itself may be under the root sign. What to do in this case? First you need to find the value of the expression under the root, and then extract the root from the number obtained as a result. If possible, it is better to get rid of roots in numerical expressions, replacing from with numeric values.
Example 5: The value of a numeric expression
Let's calculate the value of the expression with roots - 2 · 3 - 1 + 60 ÷ 4 3 + 3 · 2, 2 + 0, 1 · 0, 5.
First, we calculate the radical expressions.
2 3 - 1 + 60 ÷ 4 3 = - 6 - 1 + 15 3 = 8 3 = 2
2, 2 + 0, 1 0, 5 = 2, 2 + 0, 05 = 2, 25 = 1, 5.
Now you can calculate the value of the entire expression.
2 3 - 1 + 60 ÷ 4 3 + 3 2, 2 + 0, 1 0, 5 = 2 + 3 1, 5 = 6, 5
Often, finding the meaning of an expression with roots often requires first transforming the original expression. Let's explain this with one more example.
Example 6: The value of a numeric expression
What is 3 + 1 3 - 1 - 1
As you can see, we do not have the opportunity to replace the root with an exact value, which complicates the counting process. However, in this case, you can apply the abbreviated multiplication formula.
3 + 1 3 - 1 = 3 - 1 .
Thus:
3 + 1 3 - 1 - 1 = 3 - 1 - 1 = 1 .
Expressions with powers
If an expression contains powers, their values must be calculated before proceeding with all other actions. It happens that the exponent or the base of the degree itself are expressions. In this case, the value of these expressions is first calculated, and then the value of the degree.
Example 7: The value of a numeric expression
Let's find the value of the expression 2 3 · 4 - 10 + 16 1 - 1 2 3, 5 - 2 · 1 4.
Let's start calculating in order.
2 3 4 - 10 = 2 12 - 10 = 2 2 = 4
16 · 1 - 1 2 3, 5 - 2 · 1 4 = 16 * 0, 5 3 = 16 · 1 8 = 2.
All that remains is to perform the addition operation and find out the meaning of the expression:
2 3 4 - 10 + 16 1 - 1 2 3, 5 - 2 1 4 = 4 + 2 = 6.
It is also often advisable to simplify an expression using the properties of a degree.
Example 8: The value of a numeric expression
Let's calculate the value of the following expression: 2 - 2 5 · 4 5 - 1 + 3 1 3 6 .
The exponents are again such that their exact numerical values cannot be obtained. Let's simplify the original expression to find its value.
2 - 2 5 4 5 - 1 + 3 1 3 6 = 2 - 2 5 2 2 5 - 1 + 3 1 3 6
2 - 2 5 2 2 5 - 1 + 3 1 3 6 = 2 - 2 5 2 2 5 - 2 + 3 2 = 2 2 5 - 2 - 2 5 + 3 2
2 2 5 - 2 - 2 5 + 3 2 = 2 - 2 + 3 = 1 4 + 3 = 3 1 4
Expressions with fractions
If an expression contains fractions, then when calculating such an expression, all fractions in it must be represented as ordinary fractions and their values calculated.
If the numerator and denominator of a fraction contain expressions, then the values of these expressions are first calculated, and the final value of the fraction itself is written down. Arithmetic operations are performed in the standard order. Let's look at the example solution.
Example 9: The value of a numeric expression
Let's find the value of the expression containing fractions: 3, 2 2 - 3 · 7 - 2 · 3 6 ÷ 1 + 2 + 3 9 - 6 ÷ 2.
As you can see, there are three fractions in the original expression. Let's first calculate their values.
3, 2 2 = 3, 2 ÷ 2 = 1, 6
7 - 2 3 6 = 7 - 6 6 = 1 6
1 + 2 + 3 9 - 6 ÷ 2 = 1 + 2 + 3 9 - 3 = 6 6 = 1.
Let's rewrite our expression and calculate its value:
1, 6 - 3 1 6 ÷ 1 = 1, 6 - 0, 5 ÷ 1 = 1, 1
Often when finding the meaning of expressions, it is convenient to reduce fractions. There is an unspoken rule: before finding its value, it is best to simplify any expression to the maximum, reducing all calculations to the simplest cases.
Example 10: The value of a numeric expression
Let's calculate the expression 2 5 - 1 - 2 5 - 7 4 - 3.
We cannot completely extract the root of five, but we can simplify the original expression through transformations.
2 5 - 1 = 2 5 + 1 5 - 1 5 + 1 = 2 5 + 1 5 - 1 = 2 5 + 2 4
The original expression takes the form:
2 5 - 1 - 2 5 - 7 4 - 3 = 2 5 + 2 4 - 2 5 - 7 4 - 3 .
Let's calculate the value of this expression:
2 5 + 2 4 - 2 5 - 7 4 - 3 = 2 5 + 2 - 2 5 + 7 4 - 3 = 9 4 - 3 = - 3 4 .
Expressions with logarithms
When logarithms are present in an expression, their value is calculated from the beginning, if possible. For example, in the expression log 2 4 + 2 · 4, you can immediately write down the value of this logarithm instead of log 2 4, and then perform all the actions. We get: log 2 4 + 2 4 = 2 + 2 4 = 2 + 8 = 10.
Numerical expressions can also be found under the logarithm sign itself and at its base. In this case, the first thing to do is find their meanings. Let's take the expression log 5 - 6 ÷ 3 5 2 + 2 + 7. We have:
log 5 - 6 ÷ 3 5 2 + 2 + 7 = log 3 27 + 7 = 3 + 7 = 10.
If it is impossible to calculate the exact value of the logarithm, simplifying the expression helps to find its value.
Example 11: The value of a numeric expression
Let's find the value of the expression log 2 log 2 256 + log 6 2 + log 6 3 + log 5 729 log 0, 2 27.
log 2 log 2 256 = log 2 8 = 3 .
By the property of logarithms:
log 6 2 + log 6 3 = log 6 (2 3) = log 6 6 = 1.
Using the properties of logarithms again, for the last fraction in the expression we get:
log 5 729 log 0, 2 27 = log 5 729 log 1 5 27 = log 5 729 - log 5 27 = - log 27 729 = - log 27 27 2 = - 2.
Now you can proceed to calculating the value of the original expression.
log 2 log 2 256 + log 6 2 + log 6 3 + log 5 729 log 0, 2 27 = 3 + 1 + - 2 = 2.
Expressions with trigonometric functions
It happens that the expression contains the trigonometric functions of sine, cosine, tangent and cotangent, as well as their inverse functions. The value is calculated from before all other arithmetic operations are performed. Otherwise, the expression is simplified.
Example 12: The value of a numeric expression
Find the value of the expression: t g 2 4 π 3 - sin - 5 π 2 + cosπ.
First, we calculate the values of the trigonometric functions included in the expression.
sin - 5 π 2 = - 1
We substitute the values into the expression and calculate its value:
t g 2 4 π 3 - sin - 5 π 2 + cosπ = 3 2 - (- 1) + (- 1) = 3 + 1 - 1 = 3.
The expression value has been found.
Often in order to find the meaning of an expression with trigonometric functions, it must first be converted. Let's explain with an example.
Example 13: The value of a numeric expression
We need to find the value of the expression cos 2 π 8 - sin 2 π 8 cos 5 π 36 cos π 9 - sin 5 π 36 sin π 9 - 1.
For conversion we will use trigonometric formulas cosine of the double angle and cosine of the sum.
cos 2 π 8 - sin 2 π 8 cos 5 π 36 cos π 9 - sin 5 π 36 sin π 9 - 1 = cos 2 π 8 cos 5 π 36 + π 9 - 1 = cos π 4 cos π 4 - 1 = 1 - 1 = 0 .
General case of a numeric expression
In general trigonometric expression may contain all the elements described above: brackets, powers, roots, logarithms, functions. Let's formulate general rule finding the meanings of such expressions.
How to find the value of an expression
- Roots, powers, logarithms, etc. are replaced by their values.
- The actions in parentheses are performed.
- The remaining actions are performed in order from left to right. First - multiplication and division, then - addition and subtraction.
Let's look at an example.
Example 14: The value of a numeric expression
Let's calculate the value of the expression - 2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 ln e 2 + 1 + 3 9.
The expression is quite complex and cumbersome. It was not by chance that we chose just such an example, having tried to fit into it all the cases described above. How to find the meaning of such an expression?
It is known that when calculating the value of a complex fractional form, the values of the numerator and denominator of the fraction are first found separately, respectively. We will sequentially transform and simplify this expression.
First of all, let's calculate the value of the radical expression 2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3. To do this, you need to find the value of the sine and the expression that is the argument of the trigonometric function.
π 6 + 2 2 π 5 + 3 π 5 = π 6 + 2 2 π + 3 π 5 = π 6 + 2 5 π 5 = π 6 + 2 π
Now you can find out the value of the sine:
sin π 6 + 2 2 π 5 + 3 π 5 = sin π 6 + 2 π = sin π 6 = 1 2.
We calculate the value of the radical expression:
2 sin π 6 + 2 2 π 5 + 3 π 5 + 3 = 2 1 2 + 3 = 4
2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 = 4 = 2.
With the denominator of the fraction everything is simpler:
Now we can write the value of the entire fraction:
2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 ln e 2 = 2 2 = 1 .
Taking this into account, we write the entire expression:
1 + 1 + 3 9 = - 1 + 1 + 3 3 = - 1 + 1 + 27 = 27 .
Final result:
2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 ln e 2 + 1 + 3 9 = 27.
In this case, we were able to calculate the exact values of roots, logarithms, sines, etc. If this is not possible, you can try to get rid of them through mathematical transformations.
Calculating expression values using rational methods
Numeric values must be calculated consistently and accurately. This process can be rationalized and accelerated using various properties of operations with numbers. For example, it is known that a product is equal to zero if at least one of the factors is equal to zero. Taking this property into account, we can immediately say that the expression 2 386 + 5 + 589 4 1 - sin 3 π 4 0 is equal to zero. At the same time, it is not at all necessary to perform the actions in the order described in the article above.
It is also convenient to use the property of subtracting equal numbers. Without performing any actions, you can order that the value of the expression 56 + 8 - 3, 789 ln e 2 - 56 + 8 - 3, 789 ln e 2 is also zero.
Another technique to speed up the process is the use of identity transformations such as grouping terms and factors and placing the common factor out of brackets. 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 operations in parentheses, but by reducing the fraction, we can say that the value of the expression is 1 3 .
Finding the values of expressions with variables
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: Value of an Expression with Variables
Calculate the value of the expression 0, 5 x - y given x = 2, 4 and y = 5.
We substitute the values of the variables into the expression and calculate:
0.5 x - y = 0.5 2.4 - 5 = 1.2 - 5 = - 3.8.
Sometimes you can transform an expression so that you get its value regardless of the values of the letters and variables included in it. To do this, you need to get rid of letters and variables in the expression, if possible, using identical transformations, properties of arithmetic operations and all possible other methods.
For example, the expression x + 3 - x obviously has the value 3, and to calculate this value it is not necessary to know the value of the variable x. The value of this expression is equal to three for all values of the variable x from its range of permissible values.
One more example. The value of the expression x x is equal to one for all positive x's.
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In the 7th grade algebra course, we dealt with transformations of integer expressions, that is, expressions made up of numbers and variables using the operations of addition, subtraction and multiplication, as well as division by a number other than zero. So, the expressions are integers
In contrast, the expressions
in addition to the actions of addition, subtraction and multiplication, they contain division into an expression with variables. Such expressions are called fractional expressions.
Integer and fractional expressions are called rational expressions.
An entire expression makes sense for any values of the variables included in it, since to find the value of an entire expression you need to perform actions that are always possible.
A fractional expression may not make sense for some variable values. For example, the expression - does not make sense when a = 0. For all other values of a, this expression makes sense. The expression makes sense for those values of x and y when x ≠ y.
The values of the variables for which the expression makes sense are called acceptable values variables.
An expression of the form is known as a fraction.
A fraction whose numerator and denominator are polynomials is called a rational fraction.
Examples of rational fractions are the fractions
In a rational fraction, acceptable values of the variables are those for which the denominator of the fraction does not vanish.
Example 1. Let's find the acceptable values of the variable in the fraction
Solution To find at what values of a the denominator of the fraction becomes zero, you need to solve the equation a(a - 9) = 0. This equation has two roots: 0 and 9. Therefore, all numbers except 0 and 9 are valid values for the variable a.
Example 2. At what value of x is the value of the fraction 
equal to zero?
Solution A fraction is zero if and only if a - 0 and b ≠ 0.