Answer: _________
2. The product cost 3200 rubles. How much did this product cost after the price was reduced by 5%?
A. 3040 rub. B. 304 p. V. 1600 rub. G. 3100 p.
3. On average, students in the class completed 7.5 tasks from the proposed test. Maxim completed 9 tasks. By what percentage is his result above average?
Answer: _________
4. The row consists of natural numbers. Which of the following statistics cannot be expressed as a fraction?
A. Arithmetic mean
B. Fashion
B. Median
D. There is no such characteristic among the data.
5. Which of the equations has no roots?
A. x =x B. x =6 C. x =0 D. x =−5
6. The numbers A and B are marked on the coordinate line (Fig. 35). Compare numbers –A and B.
A. –A< В
B. –A > B
B. –A = B
D. It is impossible to compare
7. Simplify the expression a (a – 2) – (a – 1)(a + 1).
Answer: _________
8. The values of which variables need to be known in order to find the value of the expression (5a – 2b)(5a + 2b) – 4b (3a – b) + 6a (2b – 1)?
A. a and b B. a C. b
D. The value of the expression does not depend on the values of the variables
9. Solve the equation (x – 2)2 + 8x = (x – 1)(1 + x).
Answer: _________
10. Solve the system of equations ( 3x−2y=5, 5x+6y=27.
Answer: _________
11. In a 3-hour car ride and a 4-hour train ride, the tourists traveled 620 km, and the speed of the train was 10 km/h greater than the speed of the car. What is the speed of the train and the speed of the car?
Denoting the speed of the car by x km/h and the speed of the train by y km/h, we created systems of equations. Which one is composed correctly?
A. ( 3x+4y=620, x−y=10 B. ( 3x+4y=620, y−x=10
V. ( 4x+3y=620, x−y=10 G. ( 4x+3y=620, y−x=10
12. Which point does not belong to the graph of the function y = –0.6x + 1?
A. (3; –0.8) B. (–3; 0.8) B. (2; –0.2) D. (–2; 2.2)
13. In which coordinate quadrant is there not a single point on the graph of the function y = –0.6x + 1.5?
Answer: _________
14. Use the formula to define a linear function whose graph intersects the x-axis at the point (2; 0) and the y-axis at the point (0; 7).
Answer: _________ Help
A. 3040 rub. B. 304 p. V. 1600 rub. G. 3100 p. 3. On average, students in the class completed 7.5 tasks from the proposed test. Maxim completed 9 tasks. By what percentage is his result above average? Answer: _________ 4. The series consists of natural numbers. Which of the following statistics cannot be expressed as a fraction? A. Arithmetic mean B. Mode C. Median D. There is no such characteristic among the data 5. Which of the equations has no roots? A. x =x B. x =6 C. x =0 D. x =−5 6. The numbers A and B are marked on the coordinate line (Fig. 35). Compare the numbers –A and B.A. –A< В Б. –А >B B. –A = B D. Cannot be compared 7. Simplify the expression a (a – 2) – (a – 1)(a + 1). Answer: _________ 8. The values of what variables do you need to know to find the value of the expression (5a – 2b)(5a + 2b) – 4b (3a – b) + 6a (2b – 1)? A. a and b B. a C. b D. The value of the expression does not depend on the values of the variables 9. Solve the equation (x – 2)2 + 8x = (x – 1)(1 + x). Answer: _________ 10. Solve the system of equations ( 3x−2y=5, 5x+6y=27. Answer: _________ 11. In a 3-hour car ride and a 4-hour train ride, tourists traveled 620 km, and the train speed was 10 km /h is greater than the speed of the car. What is the speed of the train and the speed of the car? Having denoted by x km/h the speed of the car and by y km/h the speed of the train, we compiled systems of equations. Which of them is composed correctly? A. ( 3x+4y=620, x −y=10 B. ( 3x+4y=620, y−x=10 V. ( 4x+3y=620, x−y=10 G. ( 4x+3y=620, y−x=10 12. Which one points does not belong to the graph of the function y = –0.6x + 1? A. (3; –0.8) B. (–3; 0.8) B. (2; –0.2) D. (–2; 2,2) 13. In which coordinate quadrant is there not a single point on the graph of the function y = –0.6x + 1.5? Answer: _________ 14. Use the formula to define a linear function whose graph intersects the x-axis at the point (2; 0) and y axis at point (0; 7). Answer: _________ Option 2 1. Find the value of the expression x x−2 if x = 2.25. Answer: _________ 2. The product cost 1600 rubles. How much did the product cost after the price increased by 5 %? A. 1760 rub. B. 1700 rub. V. 1605 rub. G. 1680 rub. 3. During a shift, the shop’s turners processed an average of 12.5 parts. Petrov processed 15 parts during this shift. By what percentage is his result above average? Answer: ____________ 4. In the data series, all numbers are integers. Which of the following characteristics cannot be expressed as a fraction? A. Arithmetic mean B. Mode C. Median D. There is no such characteristic among the data 5. Which of the equations has no roots? A. x =0 B. x =7 C. x =−x D. x =−6 6. The numbers B and C are marked on the coordinate line (Fig. 36). Compare the numbers B and –C. A. B > –C B. B< –С В. В = –С Г. Сравнить невозможно 7. Упростите выражение х (х – 6) – (х – 2)(х + 2). Ответ: ___________ 8. Значения каких переменных надо знать, чтобы найти значение выражения (3х – 4у)(3х + 4у) – 3х (3х – у) + 3у (1 – х)? А. x Б. у В. x и у Г. Значение выражения не зависит от значений переменных 9. Решите уравнение (х + 3)2 – х = (х – 2)(2 + x). Ответ: ___________ 10. Решите систему уравнений { 2x+5y=−1, 3x−2y=8. Ответ: ___________ 11. Масса 5 см3 железа и 10 см3 меди равна 122 г. Масса 4 см3 железа more mass 2 cm3 of copper per 14.6 g. What is the density of iron and the density of copper? Denoting the density of iron by x g/cm3 and the density of copper by y g/cm3, we compiled systems of equations. Which system is designed correctly? A. ( 5x+10y=122, 4x−2y=14.6 B. ( 5x+10y=122, 4y−2x=14.6 C. ( 10x+5y=122, 4x−2y=14.6 D. ( 10x+5y=122, 4y−2x=14.6 12. Which point does not belong to the graph of the function y = –1.2x – 1.4? A. (–1; –0.2) B. (–2 ; 1) B. (0; –1.4) D. (–3; 2.2) 13. In which coordinate quadrant is there not a single point on the graph of the function y = 1.8x – 7.2? Answer: ___________ 14. Use a formula to define a linear function whose graph intersects the x-axis at the point (–4; 0) and the y-axis at the point (0; 3).Answer: ____________ I HAVE A FINAL TOMORROW PLEASE
Numeric expression– this is any record of numbers, arithmetic symbols and parentheses. A numerical expression can simply consist of one number. Recall that the basic 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, the recording of numbers and arithmetic symbols must be meaningful. So, for example, such an entry 5: + ∙ cannot be called a numeric expression, since it is a random set of symbols 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 the numerical expression, then as a result we will get a number. This number is called the value of a numeric expression.
Let's try to calculate what we will get as a result of performing the actions of our example. According to the order in which arithmetic operations are performed, we first perform the multiplication operation. Multiply 8 by 9. We get 72. Now add 72 and 5. We get 77.
So, 77 - meaning numerical expression 5 + 8 ∙ 9.
Numerical equality.
You can write it this way: 5 + 8 ∙ 9 = 77. Here we used the “=” sign (“Equals”) for the first time. Such a notation in which two numeric 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 – correct equality.
If we write 5 + 8 ∙ 9 = 100, then this 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 numerical expression we can also use parentheses. Parentheses affect the order in which actions are performed. So, for example, let's modify our example by adding parentheses: (5 + 8) ∙ 9. Now you first 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 correctly read an expression, you need to determine which action is performed last to calculate the value of a given numeric expression. So, if the last action is subtraction, then the expression is called “difference”. Accordingly, if the last action is sum - “sum”, division – “quotient”, multiplication – “product”, exponentiation – “power”.
For example, the numerical expression (1+5)(10-3) reads like this: “the product of the sum of the numbers 1 and 5 and the difference of the numbers 10 and 3.”
Examples of numeric expressions.
Here is an example of a more complex numerical expression:
\[\left(\frac(1)(4)+3.75 \right):\frac(1.25+3.47+4.75-1.47)(4\centerdot 0.5)\]
This numerical expression uses prime numbers, common fractions and decimals. Addition, subtraction, multiplication and division signs are also used. The fraction line also replaces the division sign. Despite the apparent complexity, finding the value of this numerical expression is quite simple. The main thing is to be able to perform operations with fractions, as well as carefully and accurately make calculations, observing the order in which the actions are performed.
In brackets we have the expression $\frac(1)(4)+3.75$ . Let's transform decimal 3.75 in ordinary.
$3.75=3\frac(75)(100)=3\frac(3)(4)$
So, $\frac(1)(4)+3.75=\frac(1)(4)+3\frac(3)(4)=4$
Next, 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 commutative law of addition, which states: “The sum does not change by changing the 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 do numerical expressions make no sense?
Let's look at 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 meaning. Numerical expressions that have no meaning are said to have “no meaning.”
If we use letters in addition to numbers in a numerical expression, then we will get an algebraic expression.
Publication date: 08/30/2014 10:58 UTC
- Geometry, a workbook for the book by Balayan E.N. "Geometry. Tasks on ready-made drawings for preparation for the Unified State Exam and Unified State Exam: grades 7-9", 7th grade, Balayan E.N., 2019
- Geometry simulator, 7th grade, for the textbook by Atanasyan L.S. and others. “Geometry. 7-9 grades", Federal State Educational Standard, Glazkov Yu.A., Egupova M.V., 2019
I. Expressions in which numbers, arithmetic symbols and parentheses can be used along with letters are called algebraic expressions.
Examples of algebraic expressions:
2m -n; 3 · (2a + b); 0.24x; 0.3a -b · (4a + 2b); a 2 – 2ab;
Since a letter in an algebraic expression can be replaced by some different numbers, then the letter is called a variable, and the algebraic expression itself is called an expression with a variable.
II. If in an algebraic expression the letters (variables) are replaced by their values and the specified actions are performed, then the resulting number is called the value of the algebraic expression.
Examples. Find the meaning of the expression:
1) a + 2b -c with a = -2; b = 10; c = -3.5.
2) |x| + |y| -|z| at x = -8; y = -5; z = 6.
Solution.
1) a + 2b -c with a = -2; b = 10; c = -3.5. Instead of variables, let's substitute their values. We get:
— 2+ 2 · 10- (-3,5) = -2 + 20 +3,5 = 18 + 3,5 = 21,5.
2) |x| + |y| -|z| at x = -8; y = -5; z = 6. Substitute the indicated values. We remember that the modulus of a negative number is equal to its opposite number, and the modulus of a positive number is equal to this number itself. We get:
|-8| + |-5| -|6| = 8 + 5 -6 = 7.
III. The values of the letter (variable) for which the algebraic expression makes sense are called the permissible values of the letter (variable).
Examples. For what values of the variable does the expression make no sense?
Solution. We know that you cannot divide by zero, therefore, each of these expressions will not make sense given the value of the letter (variable) that turns the denominator of the fraction to zero!
In example 1) this value is a = 0. Indeed, if you substitute 0 instead of a, then you will need to divide the number 6 by 0, but this cannot be done. Answer: expression 1) does not make sense when a = 0.
In example 2) the denominator of x is 4 = 0 at x = 4, therefore, this value x = 4 cannot be taken. Answer: expression 2) does not make sense when x = 4.
In example 3) the denominator is x + 2 = 0 when x = -2. Answer: expression 3) does not make sense when x = -2.
In example 4) the denominator is 5 -|x| = 0 for |x| = 5. And since |5| = 5 and |-5| = 5, then you cannot take x = 5 and x = -5. Answer: expression 4) does not make sense at x = -5 and at x = 5.
IV. Two expressions are said to be identically equal if, for any admissible values of the variables, the corresponding values of these expressions are equal.
Example: 5 (a – b) and 5a – 5b are also equal, since the equality 5 (a – b) = 5a – 5b will be true for any values of a and b. The equality 5 (a – b) = 5a – 5b is an identity.
Identity is an equality that is valid for all permissible values of the variables included in it. Examples of identities already known to you are, for example, the properties of addition and multiplication, and the distributive property.
Replacing one expression with another identically equal expression is called an identity transformation or simply a transformation of an expression. Identical transformations of expressions with variables are performed based on the properties of operations on numbers.
Examples.
a) convert the expression to identically equal using the distributive property of multiplication:
1) 10·(1.2x + 2.3y); 2) 1.5·(a -2b + 4c); 3) a·(6m -2n + k).
Solution. Let us recall the distributive property (law) of multiplication:
(a+b)c=ac+bc(distributive law of multiplication relative to addition: in order to multiply the sum of two numbers by a third number, you can multiply each term by this number and add the resulting results).
(a-b) c=a c-b c(distributive law of multiplication relative to subtraction: in order to multiply the difference of two numbers by a third number, you can multiply the minuend and subtract by this number separately and subtract the second from the first result).
1) 10·(1.2x + 2.3y) = 10 · 1.2x + 10 · 2.3y = 12x + 23y.
2) 1.5·(a -2b + 4c) = 1.5a -3b + 6c.
3) a·(6m -2n + k) = 6am -2an +ak.
b) transform the expression into identically equal, using the commutative and associative properties (laws) of addition:
4) x + 4.5 +2x + 6.5; 5) (3a + 2.1) + 7.8; 6) 5.4s -3 -2.5 -2.3s.
Solution. Let's apply the laws (properties) of addition:
a+b=b+a(commutative: rearranging the terms does not change the sum).
(a+b)+c=a+(b+c)(combinative: in order to add a third number to the sum of two terms, you can add the sum of the second and third to the first number).
4) x + 4.5 +2x + 6.5 = (x + 2x) + (4.5 + 6.5) = 3x + 11.
5) (3a + 2.1) + 7.8 = 3a + (2.1 + 7.8) = 3a + 9.9.
6) 6) 5.4s -3 -2.5 -2.3s = (5.4s -2.3s) + (-3 -2.5) = 3.1s -5.5.
V) Convert the expression to identically equal using the commutative and associative properties (laws) of multiplication:
7) 4 · X · (-2,5); 8) -3,5 · 2у · (-1); 9) 3a · (-3) · 2s.
Solution. Let's apply the laws (properties) of multiplication:
a·b=b·a(commutative: rearranging the factors does not change the product).
(a b) c=a (b c)(combinative: to multiply the product of two numbers by a third number, you can multiply the first number by the product of the second and third).
7) 4 · X · (-2,5) = -4 · 2,5 · x = -10x.
8) -3,5 · 2у · (-1) = 7у.
9) 3a · (-3) · 2c = -18ac.
If an algebraic expression is given in the form of a reducible fraction, then using the rule for reducing a fraction it can be simplified, i.e. replace it with an identically equal simpler expression.
Examples. Simplify using fraction reduction. 
Solution. To reduce a fraction means to divide its numerator and denominator by the same number (expression), other than zero. Fraction 10) will be reduced by 3b; fraction 11) will be reduced by A and fraction 12) will be reduced by 7n. We get:
Algebraic expressions are used to create formulas.
A formula is an algebraic expression written as an equality and expressing the relationship between two or more variables. Example: path formula you know s=v t(s - distance traveled, v - speed, t - time). Remember what other formulas you know.
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Now that we have learned how to add and multiply individual fractions, we can look at more complex designs. For example, what if the same problem involves adding, subtracting, and multiplying fractions?
First of all, you need to convert all fractions to improper ones. Then we perform the required actions sequentially - in the same order as for ordinary numbers. Namely:
- Exponentiation is done first - get rid of all expressions containing exponents;
- Then - division and multiplication;
- The last step is addition and subtraction.
Of course, if there are parentheses in the expression, the order of operations changes - everything that is inside the parentheses must be counted first. And remember about improper fractions: you need to highlight the whole part only when all other actions have already been completed.
Let's convert all the fractions from the first expression to improper ones, and then perform the following steps:
Now let's find the value of the second expression. There are no fractions with an integer part, but there are parentheses, so first we perform addition, and only then division. Note that 14 = 7 · 2. Then:
Finally, consider the third example. There are brackets and a degree here - it is better to count them separately. Considering that 9 = 3 3, we have:
Pay attention to the last example. To raise a fraction to a power, you must separately raise the numerator to this power, and separately, the denominator.
You can decide differently. If we recall the definition of a degree, the problem will be reduced to the usual multiplication of fractions:
Multistory fractions
Until now, we have considered only “pure” fractions, when the numerator and denominator are ordinary numbers. This is quite consistent with the definition of a number fraction given in the very first lesson.
But what if you put a more complex object in the numerator or denominator? For example, another numerical fraction? Such constructions arise quite often, especially when working with long expressions. Here are a couple of examples:
There is only one rule for working with multi-story fractions: you must get rid of them immediately. Removing “extra” floors is quite simple, if you remember that the slash means the standard division operation. Therefore, any fraction can be rewritten as follows:
Using this fact and following the procedure, we can easily reduce any multi-story fraction to an ordinary one. Take a look at the examples:
Task. Convert multistory fractions to ordinary ones:
In each case, we rewrite the main fraction, replacing the dividing line with a division sign. Also remember that any integer can be represented as a fraction with a denominator of 1. That is 12 = 12/1; 3 = 3/1. We get:
In the last example, the fractions were canceled before the final multiplication.
Specifics of working with multi-level fractions
There is one subtlety in multi-level fractions that must always be remembered, otherwise you can get the wrong answer, even if all the calculations were correct. Take a look:
- The numerator contains the single number 7, and the denominator contains the fraction 12/5;
- The numerator contains the fraction 7/12, and the denominator contains the separate number 5.
So, for one recording we got two completely different interpretations. If you count, the answers will also be different:
To ensure that the record is always read unambiguously, use a simple rule: the dividing line of the main fraction must be longer than the line of the nested fraction. Preferably several times.
If you follow this rule, then the above fractions should be written as follows:
Yes, it's probably unsightly and takes up too much space. But you will count correctly. Finally, a couple of examples where multi-story fractions actually arise:
Task. Find the meanings of the expressions:
So, let's work with the first example. Let's convert all fractions to improper ones, and then perform addition and division operations:
Let's do the same with the second example. Let's convert all fractions to improper ones and perform the required operations. In order not to bore the reader, I will omit some obvious calculations. We have:
Due to the fact that the numerator and denominator of the basic fractions contain sums, the rule for writing multi-story fractions is observed automatically. Also, in the last example, we intentionally left 46/1 in fraction form to perform division.
I will also note that in both examples the fraction bar actually replaces the parentheses: first of all, we found the sum, and only then the quotient.
Some will say that the transition to improper fractions in the second example was clearly redundant. Perhaps this is true. But by doing this we insure ourselves against mistakes, because next time the example may turn out to be much more complicated. Choose for yourself what is more important: speed or reliability.
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 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 the 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 meanings: .
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 unable to replace the root of three with its exact value, which prevents us from calculating 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, exact values You won't be able to get degrees. 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 meaning of an expression like this, fractions other than fractions should be replaced with their values before proceeding with 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 rewrite the base of the logarithm in the denominator in the form common 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 under the sign trigonometric functions If there are numerical expressions, their values are first calculated, after which the values of 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 case a numerical expression can contain roots, powers, fractions, any functions, and parentheses. Finding the values of such expressions consists of doing next steps:
- 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 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 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 property of subtracting 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 take 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're talking about 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 .
In conclusion, we note that sometimes performing transformations literal expressions and expressions with variables allows you to get 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 1 for all positive values 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.
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- 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.
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