Numerical expressions are made up of numbers, arithmetic symbols, and parentheses. If such an expression contains variables, it will be called algebraic. A trigonometric expression is an expression in which a variable is contained under the signs of trigonometric functions. Problems involving determining the values of numerical, trigonometric, and algebraic expressions are often found in school course mathematics.
Instructions
To find the value of a numeric expression, determine the order of operations in given example. For convenience, mark it with a pencil above the corresponding signs. Perform all the indicated actions in a certain order: actions in parentheses, exponentiation, multiplication, division, addition, subtraction. The resulting number will be the value of the numerical expression.
Example. Find the value of the expression (34 10+(489–296) 8):4–410. Determine the course of action. Perform the first action in the inner brackets 489–296=193. Then, multiply 193 8=1544 and 34 10=340. Next action: 340+1544=1884. Next, divide 1884:4=461 and then subtract 461–410=60. You have found the meaning of this expression.
To find the value of a trigonometric expression for a known angle?, first. To do this, use the appropriate trigonometric formulas. Calculate the given values of trigonometric functions and substitute them into the example. Follow the steps.
Example. Find the meaning of the expression 2sin 30? cos 30? tg 30? ctg 30?. Simplify this expression. To do this, use the formula tg? ctg ?=1. Get: 2sin 30? cos 30? 1=2sin 30? cos 30?. It is known that sin 30?=1/2 and cos 30?=?3/2. Therefore, 2sin 30? cos 30?=2 1/2 ?3/2=?3/2. You have found the meaning of this expression.
The meaning of an algebraic expression depends on the value of the variable. To find the value of an algebraic expression given the variables, simplify the expression. Substitute for variables certain values. Complete the necessary steps. As a result, you will receive a number, which will be the value of the algebraic expression for the given variables.
Example. Find the value of the expression 7(a+y)–3(2a+3y) with a=21 and y=10. Simplify this expression and get: a–2y. Substitute the corresponding values of the variables and calculate: a–2y=21–2 10=1. This is the value of the expression 7(a+y)–3(2a+3y) with a=21 and y=10.
note
There are algebraic expressions that do not make sense for some values of the variables. For example, the expression x/(7–a) does not make sense if a=7, because in this case, the denominator of the fraction becomes zero.
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-level 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.
This article discusses how to find the values of mathematical expressions. Let's start with simple numerical expressions and then consider cases as their complexity increases. At the end we give an expression containing letter designations, brackets, roots, special mathematical symbols, degrees, functions, etc. As per tradition, we will provide the entire theory with abundant and detailed examples.
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How to find the value of a numeric expression?
Numerical expressions, among other things, help to describe the condition of a problem in mathematical language. In general, mathematical expressions can be either very simple, consisting of a pair of numbers and arithmetic symbols, or very complex, containing functions, powers, roots, parentheses, etc. As part of a task, it is often necessary to find the meaning of a particular expression. How to do this will be discussed below.
The simplest cases
These are cases where the expression contains nothing but numbers and arithmetic operations. To successfully find the values of such expressions, you will need knowledge of the order of performing arithmetic operations without parentheses, as well as the ability to perform operations with various numbers.
If the expression contains only numbers and arithmetic signs " + " , " · " , " - " , " ÷ " , then the actions are performed from left to right in the following order: first multiplication and division, then addition and subtraction. Let's give examples.
Example 1: The value of a numeric expression
Let you need to find the values of the expression 14 - 2 · 15 ÷ 6 - 3.
Let's do the multiplication and division first. We get:
14 - 2 15 ÷ 6 - 3 = 14 - 30 ÷ 6 - 3 = 14 - 5 - 3.
Now we carry out the subtraction and get the final result:
14 - 5 - 3 = 9 - 3 = 6 .
Example 2: The value of a numeric expression
Let's calculate: 0, 5 - 2 · - 7 + 2 3 ÷ 2 3 4 · 11 12.
First we perform fraction conversion, division and multiplication:
0, 5 - 2 · - 7 + 2 3 ÷ 2 3 4 · 11 12 = 1 2 - (- 14) + 2 3 ÷ 11 4 · 11 12
1 2 - (- 14) + 2 3 ÷ 11 4 11 12 = 1 2 - (- 14) + 2 3 4 11 11 12 = 1 2 - (- 14) + 2 9.
Now let's do some addition and subtraction. Let's group the fractions and bring them to a common denominator:
1 2 - (- 14) + 2 9 = 1 2 + 14 + 2 9 = 14 + 13 18 = 14 13 18 .
The required value has been found.
Expressions with parentheses
If an expression contains parentheses, they define the order of operations in that expression. The actions in brackets are performed first, and then all the others. Let's show this with an example.
Example 3: The value of a numeric expression
Let's find the value of the expression 0.5 · (0.76 - 0.06).
The expression contains parentheses, so we first perform the subtraction operation in parentheses, and only then the multiplication.
0.5 · (0.76 - 0.06) = 0.5 · 0.7 = 0.35.
The meaning of expressions containing parentheses within parentheses is found according to the same principle.
Example 4: The value of a numeric expression
Let's calculate the value 1 + 2 1 + 2 1 + 2 1 - 1 4.
We will perform actions starting from the innermost brackets, moving to the outer ones.
1 + 2 1 + 2 1 + 2 1 - 1 4 = 1 + 2 1 + 2 1 + 2 3 4
1 + 2 1 + 2 1 + 2 3 4 = 1 + 2 1 + 2 2, 5 = 1 + 2 6 = 13.
When finding the meanings of expressions with brackets, the main thing is to follow the sequence of actions.
Expressions with roots
Mathematical expressions whose values we need to find may contain root signs. Moreover, the expression itself may be under the root sign. What to do in this case? First you need to find the value of the expression under the root, and then extract the root from the number obtained as a result. If possible, it is better to get rid of roots in numerical expressions, replacing from with numeric values.
Example 5: The value of a numeric expression
Let's calculate the value of the expression with roots - 2 · 3 - 1 + 60 ÷ 4 3 + 3 · 2, 2 + 0, 1 · 0, 5.
First, we calculate the radical expressions.
2 3 - 1 + 60 ÷ 4 3 = - 6 - 1 + 15 3 = 8 3 = 2
2, 2 + 0, 1 0, 5 = 2, 2 + 0, 05 = 2, 25 = 1, 5.
Now you can calculate the value of the entire expression.
2 3 - 1 + 60 ÷ 4 3 + 3 2, 2 + 0, 1 0, 5 = 2 + 3 1, 5 = 6, 5
Often, finding the meaning of an expression with roots often requires first transforming the original expression. Let's explain this with one more example.
Example 6: The value of a numeric expression
What is 3 + 1 3 - 1 - 1
As you can see, we do not have the opportunity to replace the root with an exact value, which complicates the counting process. However, in this case, you can apply the abbreviated multiplication formula.
3 + 1 3 - 1 = 3 - 1 .
Thus:
3 + 1 3 - 1 - 1 = 3 - 1 - 1 = 1 .
Expressions with powers
If an expression contains powers, their values must be calculated before proceeding with all other actions. It happens that the exponent or the base of the degree itself are expressions. In this case, the value of these expressions is first calculated, and then the value of the degree.
Example 7: The value of a numeric expression
Let's find the value of the expression 2 3 · 4 - 10 + 16 1 - 1 2 3, 5 - 2 · 1 4.
Let's start calculating in order.
2 3 4 - 10 = 2 12 - 10 = 2 2 = 4
16 · 1 - 1 2 3, 5 - 2 · 1 4 = 16 * 0, 5 3 = 16 · 1 8 = 2.
All that remains is to perform the addition operation and find out the meaning of the expression:
2 3 4 - 10 + 16 1 - 1 2 3, 5 - 2 1 4 = 4 + 2 = 6.
It is also often advisable to simplify an expression using the properties of a degree.
Example 8: The value of a numeric expression
Let's calculate the value of the following expression: 2 - 2 5 · 4 5 - 1 + 3 1 3 6 .
The exponents are again such that their exact numerical values cannot be obtained. Let's simplify the original expression to find its value.
2 - 2 5 4 5 - 1 + 3 1 3 6 = 2 - 2 5 2 2 5 - 1 + 3 1 3 6
2 - 2 5 2 2 5 - 1 + 3 1 3 6 = 2 - 2 5 2 2 5 - 2 + 3 2 = 2 2 5 - 2 - 2 5 + 3 2
2 2 5 - 2 - 2 5 + 3 2 = 2 - 2 + 3 = 1 4 + 3 = 3 1 4
Expressions with fractions
If an expression contains fractions, then when calculating such an expression, all fractions in it must be represented in the form ordinary fractions and calculate their values.
If the numerator and denominator of a fraction contain expressions, then the values of these expressions are first calculated, and the final value of the fraction itself is written down. Arithmetic operations are performed in the standard order. Let's look at the example solution.
Example 9: The value of a numeric expression
Let's find the value of the expression containing fractions: 3, 2 2 - 3 · 7 - 2 · 3 6 ÷ 1 + 2 + 3 9 - 6 ÷ 2.
As you can see, there are three fractions in the original expression. Let's first calculate their values.
3, 2 2 = 3, 2 ÷ 2 = 1, 6
7 - 2 3 6 = 7 - 6 6 = 1 6
1 + 2 + 3 9 - 6 ÷ 2 = 1 + 2 + 3 9 - 3 = 6 6 = 1.
Let's rewrite our expression and calculate its value:
1, 6 - 3 1 6 ÷ 1 = 1, 6 - 0, 5 ÷ 1 = 1, 1
Often when finding the meaning of expressions, it is convenient to reduce fractions. There is an unspoken rule: before finding its value, it is best to simplify any expression to the maximum, reducing all calculations to the simplest cases.
Example 10: The value of a numeric expression
Let's calculate the expression 2 5 - 1 - 2 5 - 7 4 - 3.
We cannot completely extract the root of five, but we can simplify the original expression through transformations.
2 5 - 1 = 2 5 + 1 5 - 1 5 + 1 = 2 5 + 1 5 - 1 = 2 5 + 2 4
The original expression takes the form:
2 5 - 1 - 2 5 - 7 4 - 3 = 2 5 + 2 4 - 2 5 - 7 4 - 3 .
Let's calculate the value of this expression:
2 5 + 2 4 - 2 5 - 7 4 - 3 = 2 5 + 2 - 2 5 + 7 4 - 3 = 9 4 - 3 = - 3 4 .
Expressions with logarithms
When logarithms are present in an expression, their value is calculated from the beginning, if possible. For example, in the expression log 2 4 + 2 · 4, you can immediately write down the value of this logarithm instead of log 2 4, and then perform all the actions. We get: log 2 4 + 2 4 = 2 + 2 4 = 2 + 8 = 10.
Numerical expressions can also be found under the logarithm sign itself and at its base. In this case, the first thing to do is find their meanings. Let's take the expression log 5 - 6 ÷ 3 5 2 + 2 + 7. We have:
log 5 - 6 ÷ 3 5 2 + 2 + 7 = log 3 27 + 7 = 3 + 7 = 10.
If it is impossible to calculate the exact value of the logarithm, simplifying the expression helps to find its value.
Example 11: The value of a numeric expression
Let's find the value of the expression log 2 log 2 256 + log 6 2 + log 6 3 + log 5 729 log 0, 2 27.
log 2 log 2 256 = log 2 8 = 3 .
By the property of logarithms:
log 6 2 + log 6 3 = log 6 (2 3) = log 6 6 = 1.
Using the properties of logarithms again, for the last fraction in the expression we get:
log 5 729 log 0, 2 27 = log 5 729 log 1 5 27 = log 5 729 - log 5 27 = - log 27 729 = - log 27 27 2 = - 2.
Now you can proceed to calculating the value of the original expression.
log 2 log 2 256 + log 6 2 + log 6 3 + log 5 729 log 0, 2 27 = 3 + 1 + - 2 = 2.
Expressions with trigonometric functions
It happens that the expression contains the trigonometric functions of sine, cosine, tangent and cotangent, as well as their inverse functions. The value is calculated from before all other arithmetic operations are performed. Otherwise, the expression is simplified.
Example 12: The value of a numeric expression
Find the value of the expression: t g 2 4 π 3 - sin - 5 π 2 + cosπ.
First, we calculate the values of the trigonometric functions included in the expression.
sin - 5 π 2 = - 1
We substitute the values into the expression and calculate its value:
t g 2 4 π 3 - sin - 5 π 2 + cosπ = 3 2 - (- 1) + (- 1) = 3 + 1 - 1 = 3.
The expression value has been found.
Often in order to find the meaning of an expression with trigonometric functions, it must first be converted. Let's explain with an example.
Example 13: The value of a numeric expression
We need to find the value of the expression cos 2 π 8 - sin 2 π 8 cos 5 π 36 cos π 9 - sin 5 π 36 sin π 9 - 1.
For the conversion we will use the trigonometric formulas for the cosine of a double angle and the cosine of a sum.
cos 2 π 8 - sin 2 π 8 cos 5 π 36 cos π 9 - sin 5 π 36 sin π 9 - 1 = cos 2 π 8 cos 5 π 36 + π 9 - 1 = cos π 4 cos π 4 - 1 = 1 - 1 = 0 .
General case of a numeric expression
IN general case a trigonometric expression can contain all the elements described above: brackets, powers, roots, logarithms, functions. Let's formulate general rule finding the meanings of such expressions.
How to find the value of an expression
- Roots, powers, logarithms, etc. are replaced by their values.
- The actions in parentheses are performed.
- The remaining actions are performed in order from left to right. First - multiplication and division, then - addition and subtraction.
Let's look at an example.
Example 14: The value of a numeric expression
Let's calculate the value of the expression - 2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 ln e 2 + 1 + 3 9.
The expression is quite complex and cumbersome. It was not by chance that we chose just such an example, having tried to fit into it all the cases described above. How to find the meaning of such an expression?
It is known that when calculating the value of a complex fractional form, the values of the numerator and denominator of the fraction are first found separately, respectively. We will sequentially transform and simplify this expression.
First of all, let's calculate the value of the radical expression 2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3. To do this, you need to find the value of the sine and the expression that is the argument of the trigonometric function.
π 6 + 2 2 π 5 + 3 π 5 = π 6 + 2 2 π + 3 π 5 = π 6 + 2 5 π 5 = π 6 + 2 π
Now you can find out the value of the sine:
sin π 6 + 2 2 π 5 + 3 π 5 = sin π 6 + 2 π = sin π 6 = 1 2.
We calculate the value of the radical expression:
2 sin π 6 + 2 2 π 5 + 3 π 5 + 3 = 2 1 2 + 3 = 4
2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 = 4 = 2.
With the denominator of the fraction everything is simpler:
Now we can write the value of the entire fraction:
2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 ln e 2 = 2 2 = 1 .
Taking this into account, we write the entire expression:
1 + 1 + 3 9 = - 1 + 1 + 3 3 = - 1 + 1 + 27 = 27 .
Final result:
2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 ln e 2 + 1 + 3 9 = 27.
In this case we were able to calculate exact values roots, logarithms, sines, etc. If this is not possible, you can try to get rid of them through mathematical transformations.
Calculating expression values using rational methods
Numeric values must be calculated consistently and accurately. This process can be streamlined and accelerated using various properties actions with numbers. For example, it is known that a product is equal to zero if at least one of the factors is equal to zero. Taking this property into account, we can immediately say that the expression 2 386 + 5 + 589 4 1 - sin 3 π 4 0 is equal to zero. At the same time, it is not at all necessary to perform the actions in the order described in the article above.
It is also convenient to use the property of subtracting equal numbers. Without performing any actions, you can order that the value of the expression 56 + 8 - 3, 789 ln e 2 - 56 + 8 - 3, 789 ln e 2 is also zero.
Another technique to speed up the process is the use of identity transformations such as grouping terms and factors and placing the common factor out of brackets. A rational approach to calculating expressions with fractions is to reduce the same expressions in the numerator and denominator.
For example, take the expression 2 3 - 1 5 + 3 289 3 4 3 2 3 - 1 5 + 3 289 3 4. Without performing the operations in parentheses, but by reducing the fraction, we can say that the value of the expression is 1 3 .
Finding the values of expressions with variables
Meaning literal expression and expressions with variables are found for specific given values of letters and variables.
Finding the values of expressions with variables
To find the value of a literal expression and an expression with variables, you need to substitute the given values of letters and variables into the original expression, and then calculate the value of the resulting numeric expression.
Example 15: Value of an Expression with Variables
Calculate the value of the expression 0, 5 x - y given x = 2, 4 and y = 5.
We substitute the values of the variables into the expression and calculate:
0.5 x - y = 0.5 2.4 - 5 = 1.2 - 5 = - 3.8.
Sometimes you can transform an expression so that you get its value regardless of the values of the letters and variables included in it. To do this, you need to get rid of letters and variables in the expression, if possible, using identity transformations, properties of arithmetic operations and all possible other methods.
For example, the expression x + 3 - x obviously has the value 3, and to calculate this value it is not necessary to know the value of the variable x. The value of this expression is equal to three for all values of the variable x from its range of permissible values.
One more example. The value of the expression x x is equal to one for all positive x's.
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You, as parents, in the process of educating your child, will more than once encounter the need for help in solving homework problems in mathematics, algebra and geometry. And one of the basic skills that you need to learn is how to find the meaning of an expression. Many people are at a dead end, because how many years have passed since we studied in grades 3-5? Much has already been forgotten, and some have not been learned. The rules of mathematical operations themselves are simple and you can easily remember them. Let's start with the very basics of what a mathematical expression is.
Expression Definition
A mathematical expression is a collection of numbers, action signs (=, +, -, *, /), brackets, and variables. Briefly, this is a formula whose value will need to be found. Such formulas are found in mathematics courses since school, and then haunt students who have chosen specialties related to the exact sciences. Mathematical expressions are divided into trigonometric, algebraic, and so on; let’s not get into the thicket.
- Do any calculations first on a draft, and then copy them into your workbook. This way you will avoid unnecessary crossings and dirt;
- Recalculate the total number of mathematical operations that will need to be performed in the expression. Please note that according to the rules, the operations in parentheses are performed first, then division and multiplication, and at the very end subtraction and addition. We recommend highlighting all the actions in pencil and putting numbers above the actions in the order in which they were performed. In this case, it will be easier for both you and your child to navigate;
- Start making calculations strictly following the order of actions. Let the child, if the calculation is simple, try to perform it in his head, but if it is difficult, then write with a pencil the number corresponding to the ordinal number of the expression and carry out the calculation in writing under the formula;
- As a rule, finding the value of a simple expression is not difficult if all calculations are carried out in accordance with the rules and in the right order. Most people encounter a problem precisely at this stage of finding the meaning of an expression, so be careful and do not make mistakes;
- Ban the calculator. The mathematical formulas and problems themselves may not be useful in your child’s life, but that is not the purpose of studying the subject. The main thing is the development of logical thinking. If you use calculators, the meaning of everything will be lost;
- Your task as a parent is not to solve problems for your child, but to help him in this, to guide him. Let him make all the calculations himself, and you make sure that he doesn’t make mistakes, explain why he needs to do it this way and not otherwise.
- Once the answer to the expression has been found, write it down after the “=” sign;
- Open last page mathematics textbook. Usually, there are answers for every exercise in the book. It doesn’t hurt to check whether everything has been calculated correctly.
Finding the meaning of an expression is, on the one hand, a simple procedure; the main thing is to remember the basic rules that we learned in the school mathematics course. However, on the other hand, when you need to help your child cope with formulas and solve problems, the issue becomes more complicated. After all, you are now not a student, but a teacher, and the education of the future Einstein rests on your shoulders.
We hope that our article helped you find the answer to the question of how to find the meaning of an expression, and you can easily figure out any formula!
