In the 7th grade algebra course, we were engaged in the transformation 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. Thus, expressions are integers
In contrast, expressions
in addition to the action of addition, subtraction and multiplication, they contain division by an expression with variables. Such expressions are called fractional expressions.
Integer and fractional expressions are called rational expressions.
An integer expression makes sense for any values of the variables included in it, since in order to find the value of an entire expression, you need to perform actions that are always possible.
A fractional expression for some values of variables may not make sense. For example, the expression - does not make sense for 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.
Variable values for which the expression makes sense are called valid variable values.
An expression of the form is called, as you know, a fraction.
A fraction whose numerator and denominator are polynomials is called a rational fraction.
Fractions are examples of rational fractions.
IN rational fraction admissible are those values of the variables for which the denominator of the fraction does not vanish.
Example 1 Let's find the valid values of the variable in the fraction
Solution To find at what values of a the denominator of the fraction vanishes, you need to solve the equation a (a - 9) \u003d 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 is 0 and b ≠ 0.
This article discusses how to find the values of mathematical expressions. Let's start with simple numerical expressions and then we will consider cases as their complexity increases. At the end, we give an expression containing letter designations, brackets, roots, special mathematical signs, degrees, functions, etc. The whole theory, according to tradition, will be provided with abundant and detailed examples.
How to find the value of a numeric expression?
Numeric expressions, among other things, help to describe the condition of the problem mathematical language. In general, mathematical expressions can be either very simple, consisting of a pair of numbers and arithmetic signs, or very complex, containing functions, degrees, roots, brackets, etc. As part of the task, it is often necessary to find the value of an expression. How to do this will be discussed below.
The simplest cases
These are cases where the expression contains nothing but numbers and arithmetic. To successfully find the values of such expressions, you will need knowledge of the order in which arithmetic operations are performed without brackets, as well as the ability to perform operations with different numbers.
If the expression contains only numbers and arithmetic signs " + " , " · " , " - " , " ÷ " , then operations are performed from left to right in the following order: first multiplication and division, then addition and subtraction. Let's give examples.
Example 1. Value numeric expression
Let it be necessary to find the values of the expression 14 - 2 · 15 ÷ 6 - 3 .
Let's do the multiplication and division first. We get:
14 - 2 15 ÷ 6 - 3 = 14 - 30 ÷ 6 - 3 = 14 - 5 - 3 .
Now we subtract and get the final result:
14 - 5 - 3 = 9 - 3 = 6 .
Example 2. The value of a numeric expression
Let's calculate: 0 , 5 - 2 - 7 + 2 3 ÷ 2 3 4 11 12 .
First, we perform the conversion of fractions, division and multiplication:
0 , 5 - 2 - 7 + 2 3 ÷ 2 3 4 11 12 = 1 2 - (- 14) + 2 3 ÷ 11 4 11 12
1 2 - (- 14) + 2 3 ÷ 11 4 11 12 = 1 2 - (- 14) + 2 3 4 11 11 12 = 1 2 - (- 14) + 2 9 .
Now let's do addition and subtraction. Let's group the fractions and bring them to a common denominator:
1 2 - (- 14) + 2 9 = 1 2 + 14 + 2 9 = 14 + 13 18 = 14 13 18 .
The desired value is found.
Expressions with brackets
If an expression contains brackets, then they determine the order of actions in this expression. First, the actions in brackets are performed, and then all the rest. Let's show this with an example.
Example 3. The value of a numeric expression
Find the value of the expression 0 . 5 · (0 . 76 - 0 . 06) .
There are brackets in the expression, so we first perform the subtraction operation in brackets, and only then the multiplication.
0.5 (0.76 - 0.06) = 0.5 0.7 = 0.35.
The value of expressions containing brackets in brackets is found according to the same principle.
Example 4. The value of a numeric expression
Let's calculate the value 1 + 2 · 1 + 2 · 1 + 2 · 1 - 1 4 .
We will perform actions starting from the innermost brackets, moving to the outer ones.
1 + 2 1 + 2 1 + 2 1 - 1 4 = 1 + 2 1 + 2 1 + 2 3 4
1 + 2 1 + 2 1 + 2 3 4 = 1 + 2 1 + 2 2 , 5 = 1 + 2 6 = 13 .
In finding the values of expressions with brackets, the main thing is to follow the sequence of actions.
Expressions with roots
Mathematical expressions whose values we need to find may contain root signs. Moreover, the expression itself can be under the sign of the root. How to be in that case? First you need to find the value of the expression under the root, and then extract the root from the resulting number. If possible, roots in numerical expressions should be better disposed of, replacing from with numerical values.
Example 5. The value of a numeric expression
Let's calculate the value of the expression with roots - 2 3 - 1 + 60 ÷ 4 3 + 3 2 , 2 + 0 , 1 0 , 5 .
First, we calculate the radical expressions.
2 3 - 1 + 60 ÷ 4 3 = - 6 - 1 + 15 3 = 8 3 = 2
2, 2 + 0, 1 0, 5 = 2, 2 + 0, 05 = 2, 25 = 1, 5.
Now we can calculate the value of the entire expression.
2 3 - 1 + 60 ÷ 4 3 + 3 2, 2 + 0, 1 0, 5 = 2 + 3 1, 5 = 6, 5
Often, to find the value of an expression with roots, it is often necessary to first transform the original expression. Let's explain this with another example.
Example 6. The value of a numeric expression
What is 3 + 1 3 - 1 - 1
As you can see, we do not have the ability to replace the root with an exact value, which complicates the counting process. However, in this case, you can apply the abbreviated multiplication formula.
3 + 1 3 - 1 = 3 - 1 .
In this way:
3 + 1 3 - 1 - 1 = 3 - 1 - 1 = 1 .
Expressions with powers
If the expression contains powers, their values must be calculated before proceeding with all other actions. It happens that the exponent itself or the base of the degree are expressions. In this case, the value of these expressions is calculated first, and then the value of the degree.
Example 7. The value of a numeric expression
Find the value of the expression 2 3 4 - 10 + 16 1 - 1 2 3 , 5 - 2 1 4 .
We begin to calculate in order.
2 3 4 - 10 = 2 12 - 10 = 2 2 = 4
16 1 - 1 2 3, 5 - 2 1 4 = 16 * 0, 5 3 = 16 1 8 = 2.
It remains only to carry out the addition operation and find out the value of the expression:
2 3 4 - 10 + 16 1 - 1 2 3 , 5 - 2 1 4 = 4 + 2 = 6 .
It is also often advisable to simplify the expression using the properties of the degree.
Example 8. The value of a numeric expression
Let's calculate the value of the following expression: 2 - 2 5 · 4 5 - 1 + 3 1 3 6 .
The exponents are again such that their exact numerical values cannot be obtained. Simplify the original expression to find its value.
2 - 2 5 4 5 - 1 + 3 1 3 6 = 2 - 2 5 2 2 5 - 1 + 3 1 3 6
2 - 2 5 2 2 5 - 1 + 3 1 3 6 = 2 - 2 5 2 2 5 - 2 + 3 2 = 2 2 5 - 2 - 2 5 + 3 2
2 2 5 - 2 - 2 5 + 3 2 = 2 - 2 + 3 = 1 4 + 3 = 3 1 4
Expressions with fractions
If an expression contains fractions, then when calculating such an expression, all fractions in it must be represented as ordinary fractions and their values \u200b\u200bcalculated.
If there are expressions in the numerator and denominator of the fraction, then the values of these expressions are first calculated, and the final value of the fraction itself is recorded. Arithmetic operations are performed in the standard order. Let's consider an example solution.
Example 9. The value of a numeric expression
Let's find the value of the expression containing fractions: 3 , 2 2 - 3 7 - 2 3 6 ÷ 1 + 2 + 3 9 - 6 ÷ 2 .
As you can see, there are three fractions in the original expression. Let us first calculate their values.
3 , 2 2 = 3 , 2 ÷ 2 = 1 , 6
7 - 2 3 6 = 7 - 6 6 = 1 6
1 + 2 + 3 9 - 6 ÷ 2 = 1 + 2 + 3 9 - 3 = 6 6 = 1 .
Let's rewrite our expression and calculate its value:
1 , 6 - 3 1 6 ÷ 1 = 1 , 6 - 0 , 5 ÷ 1 = 1 , 1
Often, when finding the values of expressions, it is convenient to reduce fractions. There is an unspoken rule: before finding its value, 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, if possible, is calculated from the very beginning. For example, in the expression log 2 4 + 2 4, you can immediately write the value of this logarithm instead of log 2 4, and then perform all the actions. We get: log 2 4 + 2 4 = 2 + 2 4 = 2 + 8 = 10 .
Numeric expressions can also be found under the sign of the logarithm and at its base. In this case, the first step is to find their values. Let's take the expression log 5 - 6 ÷ 3 5 2 + 2 + 7 . We have:
log 5 - 6 ÷ 3 5 2 + 2 + 7 = log 3 27 + 7 = 3 + 7 = 10 .
If it is impossible to calculate the exact value of the logarithm, simplifying the expression helps to find its value.
Example 11. The value of a numeric expression
Find the value of the expression log 2 log 2 256 + log 6 2 + log 6 3 + log 5 729 log 0 , 2 27 .
log 2 log 2 256 = log 2 8 = 3 .
According to the property of logarithms:
log 6 2 + log 6 3 = log 6 (2 3) = log 6 6 = 1 .
Again applying the properties of logarithms, for the last fraction in the expression we get:
log 5 729 log 0 , 2 27 = log 5 729 log 1 5 27 = log 5 729 - log 5 27 = - log 27 729 = - log 27 27 2 = - 2 .
Now you can proceed to the calculation of the value of the original expression.
log 2 log 2 256 + log 6 2 + log 6 3 + log 5 729 log 0 , 2 27 = 3 + 1 + - 2 = 2 .
Expressions with trigonometric functions
It happens that in the expression there are trigonometric functions of sine, cosine, tangent and cotangent, as well as functions that are inverse to them. From the value are calculated before all other arithmetic operations are performed. Otherwise, the expression is simplified.
Example 12. The value of a numeric expression
Find the value of the expression: t g 2 4 π 3 - sin - 5 π 2 + cosπ.
First, we calculate the values trigonometric functions included in the expression.
sin - 5 π 2 \u003d - 1
Substitute the values in the expression and calculate its value:
t g 2 4 π 3 - sin - 5 π 2 + cosπ \u003d 3 2 - (- 1) + (- 1) \u003d 3 + 1 - 1 \u003d 3.
The value of the expression is found.
Often, in order to find the value of an expression with trigonometric functions, it must first be converted. Let's explain with an example.
Example 13. The value of a numeric expression
It is necessary to find the value of the expression cos 2 π 8 - sin 2 π 8 cos 5 π 36 cos π 9 - sin 5 π 36 sin π 9 - 1.
For the transformation we will use trigonometric formulas cosine double angle and the cosine of the sum.
cos 2 π 8 - sin 2 π 8 cos 5 π 36 cos π 9 - sin 5 π 36 sin π 9 - 1 = cos 2 π 8 cos 5 π 36 + π 9 - 1 = cos π 4 cos π 4 - 1 = 1 - 1 = 0 .
General case of numeric expression
In the general case, a trigonometric expression can contain all the elements described above: brackets, degrees, roots, logarithms, functions. Let's formulate general rule finding the values 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 steps are performed in order from left to right. First - multiplication and division, then - addition and subtraction.
Let's take an example.
Example 14. The value of a numeric expression
Let's calculate what the value of the expression is - 2 sin π 6 + 2 2 π 5 + 3 π 5 + 3 ln e 2 + 1 + 3 9 .
The expression is quite complex and cumbersome. It is not by chance that we chose just such an example, trying to fit into it all the cases described above. How to find the value of such an expression?
It is known that when calculating the value of a complex fractional form, first the values of the numerator and denominator of the fraction are found separately, respectively. We will successively transform and simplify this expression.
First of all, we calculate the value of the radical expression 2 sin π 6 + 2 2 π 5 + 3 π 5 + 3. To do this, you need to find the value of the sine, and the expression that is the argument of the trigonometric function.
π 6 + 2 2 π 5 + 3 π 5 = π 6 + 2 2 π + 3 π 5 = π 6 + 2 5 π 5 = π 6 + 2 π
Now you can find out the value of the sine:
sin π 6 + 2 2 π 5 + 3 π 5 = sin π 6 + 2 π = sin π 6 = 1 2 .
We calculate the value of the radical expression:
2 sin π 6 + 2 2 π 5 + 3 π 5 + 3 = 2 1 2 + 3 = 4
2 sin π 6 + 2 2 π 5 + 3 π 5 + 3 = 4 = 2.
With the denominator of a fraction, everything is easier:
Now we can write down the value of the whole fraction:
2 sin π 6 + 2 2 π 5 + 3 π 5 + 3 ln e 2 = 2 2 = 1.
With this in mind, we write the entire expression:
1 + 1 + 3 9 = - 1 + 1 + 3 3 = - 1 + 1 + 27 = 27 .
Final Result:
2 sin π 6 + 2 2 π 5 + 3 π 5 + 3 ln e 2 + 1 + 3 9 = 27.
In this case, we were able to calculate exact values for roots, logarithms, sines, and so on. If this is not possible, you can try to get rid of them by mathematical transformations.
Computing Expressions in Rational Ways
Numeric values must be calculated consistently and accurately. This process can be rationalized and accelerated by using various properties of operations with numbers. For example, it is known that the product is equal to zero if at least one of the factors is equal to zero. Given this property, we can immediately say that the expression 2 386 + 5 + 589 4 1 - sin 3 π 4 0 is equal to zero. In this case, it is not at all necessary to perform the steps in the order described in the article above.
It is also convenient to use the property of subtracting equal numbers. Without performing any actions, it is possible to order that the value of the expression 56 + 8 - 3 , 789 ln e 2 - 56 + 8 - 3 , 789 ln e 2 is also equal to zero.
Another technique that allows you to speed up the process is the use of identical transformations such as grouping terms and factors and taking the common factor out of brackets. A rational approach to calculating expressions with fractions is to reduce the same expressions in the numerator and denominator.
For example, let's take the expression 2 3 - 1 5 + 3 289 3 4 3 2 3 - 1 5 + 3 289 3 4 . Without performing actions in brackets, but by reducing the fraction, we can say that the value of the expression is 1 3 .
Finding the values of expressions with variables
The meaning of a literal expression and an expression with variables is found for specific setpoints letters and variables.
Finding the values of expressions with variables
To find the value of a literal expression and an expression with variables, you need to substitute the given values of letters and variables into the original expression, and then calculate the value of the resulting numeric expression.
Example 15. The value of an expression with variables
Calculate the value of the expression 0 , 5 x - y given x = 2 , 4 and y = 5 .
We substitute the values of the variables into the expression and calculate:
0 . 5 x - y = 0 . 5 2 . 4 - 5 = 1 . 2 - 5 = - 3 . 8 .
Sometimes it is possible to transform an expression in such a way as to obtain its value regardless of the values of the letters and variables included in it. To do this, 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 ways.
For example, the expression x + 3 - x obviously has the value 3, and it is not necessary to know the value of x to calculate this value. Meaning given expression is equal to three for all values of variable x from its range of acceptable values.
One more example. The value of the expression x x is equal to one for all positive x's.
If you notice a mistake in the text, please highlight 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 take a look at 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, 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 equal to 6 .
Answer:
14−2 15:6−3=6 .
Example.
Find the value of the expression.
Solution.
In this example, we first need to perform the multiplication 2 (−7) and division with multiplication in the expression. Remembering how , we find 2 (−7)=−14 . And to perform actions in the expression, first 
, then 
, and execute: 
.
We substitute the obtained values into the original expression: .
But what about when there is a numeric expression under the root sign? To get the value of such a root, you must first find the value of the root expression, following the accepted order of operations. For example, .
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 value of the expression with roots.
Solution.
First, 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 convert it. Let's show an example solution.
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 difference of squares formula: . Considering , we get 
. So the value of the original expression is 1 .
Answer:
.
With degrees
If the base and exponent are numbers, then their value is calculated by the definition of the degree, for example, 3 2 =3 3=9 or 8 −1 =1/8 . There are also entries 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.
The original expression has two powers 2 3 4−10 and (1−1/2) 3.5−2 1/4 . Their values must be calculated before performing the rest of the steps.
Let's start with the power 2 3·4−10 . Its indicator contains a numeric 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 .
There are expressions in the base and exponent (1−1/2) 3.5−2 1/4, we calculate their values in order to find the value of the degree later. 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 value of an expression 
.
Solution.
Judging by the exponents in this expression, the exact values of the degrees cannot be obtained. Let's try to simplify the original expression, maybe it will help to find its value. We have 
Answer:
.
Powers in expressions often go hand in hand with logarithms, but we'll talk about finding the values of expressions with logarithms in one of.
Finding the value of an expression with fractions
Numeric expressions in their entry may contain fractions. When you want to find the value of such an expression, fractions other than ordinary fractions should be replaced by their values before performing other 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 in fact a quotient of the form (a):(b) , since .
Let's consider an example solution.
Example.
Find the value 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 and replace them with their values. Let's do it.
The numerator and denominator of a fraction are numbers. To find the value of such a fraction, we 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 numeric 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 steps: .
Answer:
.
Often, when finding the values of expressions with fractions, you have to perform simplification fractional expressions , based on the performance of actions with fractions and on the reduction of fractions.
Example.
Find the value of an expression 
.
Solution.
The root of five is not completely extracted, so to find the value of the original expression, let's simplify it first. For this get rid of the 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 given 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 other actions. For example, when finding the value of the expression log 2 4+2 3 , the logarithm of log 2 4 is replaced by its value 2 , after which the rest of the operations 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, then 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 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 its preliminary simplification using . In this case, you need to have a good command of the material of the article. transformation of 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 , hence log 2 (log 2 256)=log 2 8=log 2 2 3 =3.
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) , so log 6 2+log 6 3=log 6 (2 3)=log 6 6=1.
Now let's deal with fractions. To begin with, we rewrite the base of the logarithm in the denominator in the form common fraction as 1/5 , after which we use the properties of logarithms, which will allow us to get the value of the fraction: 
.
It remains only 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., then 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 value of an 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 should be noted that the calculation of the values of expressions with sines, cosines, etc. often requires prior transformations trigonometric expression .
Example.
What is the value of the trigonometric expression 
.
Solution.
Let's transform the original expression using , in this case we need the double angle cosine formula and the sum cosine formula: 
The transformations done helped us find the value of the expression.
Answer:
.
General case
In the general case, a numeric expression can contain roots, degrees, fractions, and any functions, and brackets. Finding the values of such expressions consists in performing the following actions:
- first roots, degrees, fractions, etc. are replaced by their values,
- further actions in parentheses,
- and in order from left to right, the remaining operations are performed - multiplication and division, followed by addition and subtraction.
The above actions are performed until the final result is obtained.
Example.
Find the value of an expression 
.
Solution.
The form of this expression is rather complicated. In this expression, we see a fraction, roots, degrees, sine and logarithm. How to find its meaning?
Moving along the record from left to right, we come across a fraction of the form 
. We know that when dealing 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 a root of the form 
. To determine its value, you must first 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 is what we can do: . Then from where and 
.
With the denominator, everything 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 roots, degrees, etc., then you can try to get rid of them using any transformations, and then return to calculating the value according to the specified scheme.
Rational Ways to Calculate Values of Expressions
Calculating the values of numerical expressions requires consistency and accuracy. Yes, it is necessary to adhere to the sequence of actions recorded in the previous paragraphs, but this should not be done blindly and mechanically. By this we mean that it is often possible to rationalize the process of finding the value of an expression. For example, some properties of actions with numbers allow you to 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 zero. If we followed the standard order of operations, then we would first have to calculate the values of cumbersome expressions in brackets, 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, then the result will be zero. This property can be considered more broadly: the difference of two identical numerical expressions is equal to zero. For example, without calculating the value of expressions in brackets, 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.
The rational calculation of the values of expressions can be facilitated by identical transformations. For example, it is useful grouping of terms and factors, not less often used taking the common factor out of brackets. 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 take much more time.
In conclusion of this paragraph, let's pay attention to the rational approach to calculating the values of expressions with fractions - the same factors in the numerator and denominator of the fraction are reduced. For example, reducing 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
Meaning of literal and variable expressions is found for specific given values of letters and variables. I.e, 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: in the original expression, you need to substitute the given values of letters or variables, 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 for x=2.4 and y=5 .
Solution.
To find the required value of the expression, you first need to substitute these variable values into the original expression, and then perform the following actions: 0.5 2.4−5=1.2−5=−3.8 .
Answer:
−3,8 .
In conclusion, we note that sometimes the transformation 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 to become 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 acceptable values (ODZ). Another example: the value of the expression is equal to 1 for all positive values x , so the range of acceptable values for the variable x in the original expression is the set of positive numbers, and equality takes place on this area.
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