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.
V this example we first need to do 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 initially for of this 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 will 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 in this case we need a double angle cosine formula and a sum cosine formula: 
The performed transformations helped us find the meaning of the expression.
Answer:
.
General case
V general case 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 recorded in previous paragraphs, but you don't 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 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 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, it comes about finding the meaning 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 an alphabetic 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 positive numbers, and equality takes place in this area.
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.
A numeric expression is the notation of numbers in conjunction with arithmetic operations and parentheses. When variables are used in an expression together with numbers and the whole expression is composed with meaning, then it is called an algebraic (literal) expression. If the expression contains direct, derivative, inverse and other trigonometric functions, then the expression is called trigonometric. A large number of examples and tasks using various expressions are detailed in school course mathematics.
The main things to remember:
1. The value of a numeric expression will be the number obtained when performing arithmetic operations in this expression. The main thing is to consistently perform arithmetic operations. For simplicity of the whole operation, actions can be numbered. If the expression contains brackets, then first of all we perform the action corresponding to the sign in brackets. Exponentiation will be the next step. Further, by priority, we perform multiplication or division, and only at the very end, addition and subtraction.
And now we will find the value of the numerical expression 5 + 20 * (60-45). First, we get rid of the parentheses. Performing the action, we get 60-45 = 15. Now we have 5 + 20 * 15. The next action is multiplication 20 * 15 = 300. And the last action will be addition, we perform it and we get the final result 5 + 300 = 305.
2. With a known angle? When working with trigonometric expressions, you need to know basic trigonometric formulas to help you simplify the expression. Find the value of the expression cos 12? cos 18? - sin 12? sin 18 ?. To simplify this expression, use the formula cos (? +?) = Cos? cos? - sin? sin?, then we get cos 12? cos 18? - sin 12? sin 18? = cos (12? +18?) = cos30? = v3? 2.
3. Expressions with variables. It must be remembered that the value of an algebraic expression directly depends on the variable. Variables can be denoted with Greek letters or Latin alphabet... When do we have set parameters algebraic expression, first you need to simplify it. After that, you need to substitute the specified variables and produce arithmetic operations... As a result, given the variables, we get a number, which will be the value of the algebraic expression. Consider an example where you need to find the value of the expression 3 (a + y) +2 (3a + 2y) with a = 4 and y = 5. Simplify this expression and get 3a + 3y + 6a + 4y = 9a + 7y. Now you need to substitute the value of the variables and calculate, the result will be the value of the expression. So, we have 9a + 7y with a = 4 and y = 5 we get 36 + 35 = 71. Note that algebraic expressions do not always make sense. For example, this expression 15: (b-4) makes sense for any b except b = 4.
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 last action 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
Formula
Addition, subtraction, multiplication, division are arithmetic operations (or arithmetic operations). These arithmetic operations correspond to the signs of arithmetic operations:
+ (read " a plus") - the sign of the addition operation,
- (read " minus") - sign subtraction operations,
∙ (read " multiply") - sign multiplication operations,
: (read " divide") is the sign of the division operation.
A record consisting of numbers connected by signs of arithmetic operations is called numerical expression. A numeric expression may also contain parentheses For example, record 1290 : 2 - (3 + 20 ∙ 15) is a numeric expression.
The result of performing actions on numbers in a numerical expression is called the value of a numeric expression... Doing this is called evaluating the value of a numeric expression. Before writing the value of a numeric expression, put equal sign"=". Table 1 shows examples of numeric expressions and their meanings.
A record consisting of numbers and small letters of the Latin alphabet, connected by signs of arithmetic operations, is called literal expression... This entry may contain parentheses. For example, the entry a +b - 3 ∙c is a literal expression. Instead of letters in a literal expression, you can substitute different numbers... In this case, the meaning of the letters can change, therefore the letters in the literal expression are also called variables.
Substituting numbers instead of letters into the literal expression and calculating the value of the resulting numerical expression, they find the value of the literal expression given the values of the letters(for the given values of the variables). Table 2 shows examples of letter expressions.
A literal expression may not matter if substitution of letter values results in a numeric expression whose value is for natural numbers could not be found. Such a numeric expression is called incorrect for natural numbers. It is also said that the meaning of such an expression “ undefined" for natural numbers, and the expression itself "Doesn't make sense"... For example, the literal expression a - b does not matter for a = 10 and b = 17. Indeed, for natural numbers, the diminished cannot be less than the subtracted. For example, having only 10 apples (a = 10), you cannot give away 17 of them (b = 17)!
Table 2 (column 2) provides an example of an alphabetic expression. Fill in the table completely by analogy.
For natural numbers, the expression 10 -17 incorrect (does not make sense), i.e. the difference 10 -17 cannot be expressed as a natural number. Another example: you cannot divide by zero, so for any natural number b, the quotient b: 0 undefined.
Mathematical laws, properties, some rules and relationships are often written in letter form (i.e. in the form of letter expression). In these cases, the literal expression is called formula... For example, if the sides of the heptagon are equal a,b,c,d,e,f,g, then the formula (literal expression) to calculate its perimeter p looks like:
p =a +b +c +d +e +f +g
For a = 1, b = 2, c = 4, d = 5, e = 5, f = 7, g = 9, the perimeter of the heptagon p = a + b + c + d + e + f + g = 1 + 2 + 4 + 5 + 5 + 7 + 9 = 33.
For a = 12, b = 5, c = 20, d = 35, e = 4, f = 40, g = 18, the perimeter of another heptagon is p = a + b + c + d + e + f + g = 12 + 5 + 20 + 35 + 4 + 40 + 18 = 134.
Block 1. Dictionary
Compile a glossary of new terms and definitions from the paragraph. To do this, write words from the list of terms below in the empty cells. In the table (at the end of the block) indicate the numbers of terms in accordance with the numbers of the frames. It is recommended to carefully review the paragraph before filling in the cells of the dictionary.
- Operations: addition, subtraction, multiplication, division.
2. Signs "+" (plus), "-" (minus), "∙" (multiply, " : " (divide).
3. A record consisting of numbers that are connected by signs of arithmetic operations and in which brackets may also be present.
4. The result of performing actions on numbers in numerical terms.
5. The sign before the value of a numeric expression.
6. A record consisting of numbers and small letters of the Latin alphabet, connected with each other by signs of arithmetic operations (brackets may also be present).
7. The general name of letters in literal expression.
8. The value of a numeric expression, which is obtained by substitution of variables.in a literal expression.
9.Numeric expression whose value for natural numbers cannot be found.
10. Numeric expression, the value of which for natural numbers can be found.
11. Mathematical laws, properties, some rules and relationships, written down in letter form.
12. The alphabet, the small letters of which are used to write alphabetic expressions.
Block 2. Establish correspondence
Establish a correspondence between the item in the left column and the solution in the right. Write the answer in the form: 1a, 2d, 3b ...
Block 3. Facet test. Numeric and literal expressions
Facet tests replace collections of problems in mathematics, but they compare favorably with them in that they can be solved on a computer, checked solutions and immediately recognized the result of the work. This test contains 70 problems. But you can solve problems by choice, for this there is an assessment table, where simple tasks and more difficult ones are indicated. Below is the test.
- Given a triangle with sides c,d,m, expressed in cm
- Given a quadrangle with sides b,c,d,m expressed in m
- Vehicle speed in km / h is b, the time of movement in hours is d
- The distance traveled by the tourist in m hours is With km
- Distance traveled by a tourist moving at speed m km / h is b km
- The sum of two numbers is 15 more than the second
- The difference is less than the reduced by 7
- The passenger liner has two decks with the same number of passenger seats. In each of the deck rows m seats, rows on the deck on n more than seats in a row
- Petya is m years old, Masha is n years old, and Katya is k years younger than Petya and Masha together
- m = 8, n = 10, k = 5
- m = 6, n = 8, k = 15
- t = 121, x = 1458
- The meaning of this expression
- The literal expression for the perimeter is
- Perimeter expressed in centimeters
- Formula for the path s covered by the car
- The formula for the speed v, the movement of the tourist
- Formula of time t, tourist movement
- Distance traveled by the car in kilometers
- Tourist speed in kilometers per hour
- Tourist travel time in hours
- The first number is ...
- The subtracted is….
- Expression for the largest number of passengers that the liner can carry for k flights
- The largest number of passengers that the liner can carry in k flights
- Letter expression for Katya's age
- Katya's age
- The coordinate of point B, if the coordinate of point C is t
- The coordinate of point D, if the coordinate of point C is t
- The coordinate of point A, if the coordinate of point C is t
- BD segment length on a number beam
- The length of the segment CA on the number beam
- The length of the segment DA on the number beam
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 signs, degree, function, 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. Generally 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. Within the framework of a task, it is often necessary to find the meaning of an expression. How to do it, and there will be a speech 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. The actions in brackets are performed first, 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. If possible, it is better to get rid of roots in numerical expressions, replacing from with numerical values.
Example 5. The value of a numeric expression
Let's calculate the value of the expression with roots - 2 3 - 1 + 60 ÷ 4 3 + 3 2, 2 + 0, 1 · 0, 5.
First, we calculate the radical expressions.
2 3 - 1 + 60 ÷ 4 3 = - 6 - 1 + 15 3 = 8 3 = 2
2, 2 + 0, 1 0, 5 = 2, 2 + 0, 05 = 2, 25 = 1, 5.
Now 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 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's formulate general rule finding the values of such expressions.
How to find the meaning of an expression
- Roots, degrees, logarithms, etc. are replaced by their values.
- 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. 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 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 of terms and factors and common factor outside the 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 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 in the original expression target values letters and variables, and then calculate the value of the resulting numeric 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.
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