Find the meaning of the expression in what expressions. Converting expressions. Detailed theory (2019)

As a rule, children start learning algebra already in the elementary grades. After mastering the basic principles of working with numbers, they solve examples with one or more unknown variables. Finding the meaning of an expression of this kind can be quite difficult, but if you simplify it using the knowledge of elementary school, everything will work out quickly and easily.

What is the meaning of an expression

A numerical expression is called algebraic notation consisting of numbers, parentheses, and signs, if it makes sense.

In other words, if it is possible to find the meaning of an expression, then the record is not devoid of meaning, and vice versa.

Examples of the following entries are valid numeric constructs:

• 3*8-2;
• 15/3+6;
• 0,3*8-4/2;
• 3/1+15/5;

A single number will also represent a numerical expression like the number 18 from the above example.
Examples of invalid numeric constructs that don't make sense:

• *7-25);
• 16/0-;
• (*-5;

Incorrect numeric examples are just a set of mathematical symbols and don't make any sense.

How to find the meaning of an expression

Since there are arithmetic signs in such examples, we can conclude that they allow you to perform arithmetic calculations. To calculate the signs or, in other words, to find the value of an expression, it is necessary to perform the corresponding arithmetic manipulations.

As an example, consider the following construction: (120-30) / 3 = 30. The number 30 will be the value of the numeric expression (120-30) / 3.

Instructions:

Numeric Equality Concept

A numerical equality is a situation when two parts of an example are separated by an "=" sign. That is, one part is completely equal (identical) to the other, even if displayed in the form of other combinations of symbols and numbers.
For example, any construction like 2 + 2 = 4 can be called a numerical equality, because even if the parts are reversed, the meaning will not change: 4 = 2 + 2. The same goes for more complex constructs such as parentheses, division, multiplication, fractions, and so on.

How to find the meaning of an expression correctly

To correctly find the value of an expression, it is necessary to perform calculations according to a certain order of actions. This order is taught even in mathematics lessons, and later in algebra lessons in primary school... It is also known as the rungs of arithmetic operations.

Arithmetic steps:

1. The first step is addition and subtraction of numbers.
2. The second stage is division and multiplication.
3. Third step - numbers are squared or cube.

Observing the following rules, you can always correctly determine the meaning of the expression:

1. Proceed from step 3 to step 1 if there are no parentheses in the example. That is, first square or cube, then divide or multiply, and only then add and subtract.
2. In constructions with brackets, first perform the actions in brackets, and then follow the above procedure. If there are more than one parenthesis, also use the procedure from the first paragraph.
3. In the examples in the form of a fraction, first find out the result in the numerator, then in the denominator, and then divide the first by the second.

Finding the meaning of the expression will not be difficult if you master the elementary knowledge of the initial courses in algebra and mathematics. Guided by the information described above, you can solve any problem, even of increased complexity.

Find out the password from VK, knowing the login

(34 ∙ 10 + (489-296) ∙ 8): 4-410. Determine the course of action. Perform the first step in the inner brackets 489-296 = 193. Then, multiply 193 ∙ 8 = 1544 and 34 ∙ 10 = 340. Next action: 340 + 1544 = 1884. Next, do the division 1884: 4 = 461 and then subtract 461-410 = 60. You have found the meaning of this expression.

Example. Find the value of the expression 2sin 30º ∙ cos 30º ∙ tg 30º ∙ ctg 30º. Simplify this expression. To do this, use the formula tg α ∙ ctg α = 1. Get: 2sin 30º ∙ cos 30º ∙ 1 = 2sin 30º ∙ cos 30º. It is known that sin 30º = 1/2 and cos 30º = √3 / 2. Therefore, 2sin 30º ∙ cos 30º = 2 ∙ 1/2 ∙ √3 / 2 = √3 / 2. You have found the meaning of this expression.

The value of an algebraic expression from. To find the value of an algebraic expression for given variables, simplify the expression. Substitute specific values ​​for variables. Take the necessary steps. As a result, you will get 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, get: a – 2y. Plug in the corresponding values ​​of the variables and calculate: a – 2y = 21–2 ∙ 10 = 1. This is the meaning 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) is meaningless if a = 7, since in this case, the denominator of the fraction vanishes.

Sources:

• find smallest value expressions
• Find the values ​​of the expressions for c 14

Learning to simplify expressions in mathematics is simply necessary in order to correctly and quickly solve problems, various equations. Simplifying an expression means fewer steps, which makes calculations easier and saves time.

Instructions

Learn to calculate degrees with. When the powers are multiplied with, numbers are obtained, the base of which is the same, and the exponents are added b ^ m + b ^ n = b ^ (m + n). When dividing powers with on the same grounds the power of a number is obtained, the base of which remains the same, and the exponents are subtracted, and the divisor exponent b ^ m is subtracted from the index of the dividend: b ^ n = b ^ (m-n). When raising a power to a power, a power of a number is obtained, the base of which remains the same, and the exponents are multiplied (b ^ m) ^ n = b ^ (mn) When raising to a power, each factor is raised to this power. (Abc) ^ m = a ^ m * b ^ m * c ^ m

Factor polynomials, i.e. think of them as the product of several factors - and monomials. Factor out the common factor. Learn basic abbreviated multiplication formulas: difference of squares, square of difference, sum, difference of cubes, cube of sum and difference. For example, m ^ 8 + 2 * m ^ 4 * n ^ 4 + n ^ 8 = (m ^ 4) ^ 2 + 2 * m ^ 4 * n ^ 4 + (n ^ 4) ^ 2. These formulas are the main ones in simplification. Use the method of selecting a complete square in a trinomial of the form ax ^ 2 + bx + c.

Reduce fractions as often as possible. For example, (2 * a ^ 2 * b) / (a ​​^ 2 * b * c) = 2 / (a ​​* c). But remember that only factors can be canceled. If the numerator and denominator of an algebraic fraction are multiplied by the same nonzero number, then the value of the fraction will not change. There are two ways to transform expressions: chain and action. The second way is preferable, because it is easier to check the results of intermediate actions.

It is often necessary to extract roots in expressions. Even roots are extracted only from non-negative expressions or numbers. Odd roots are derived from any expression.

Sources:

• simplification of power expressions

Trigonometric functions first appeared as tools for abstract mathematical calculations of the dependences of the values ​​of acute angles in right triangle from the lengths of its sides. Now they are very widely used in both scientific and technical areas human activity. For practical calculations trigonometric functions from the given arguments you can use different tools- below are some of the most accessible ones.

Instructions

Use, for example, the one installed by default along with operating system calculator program. It opens by selecting the "Calculator" item in the "System Tools" folder from the "Standard" subsection located in the "All Programs" section. This section can be opened by clicking on the "Start" button the main menu of the operating room. If you are using Windows 7, you can simply enter "Calculator" in the "Find programs and files" field of the main menu, and then click on the corresponding link in the search results.

Count the number necessary actions and think about the order in which they should be performed. If it makes you difficult this question, note that the actions enclosed in brackets are performed first, then division and multiplication; and subtraction is done last. To make it easier to remember the algorithm of the actions performed, in the expression above each action operator sign (+, -, *, :), use a thin pencil to write the numbers corresponding to the actions.

Proceed with the first step, adhering to the established order. Count in your head if the steps are easy to do verbally. If calculations are required (in a column), write them under the expression, indicating serial number actions.

Clearly track the sequence of actions performed, evaluate what to subtract from what, what to divide into what, etc. Very often, the answer in the expression turns out to be incorrect due to mistakes made at this stage.

Distinctive feature expression is the presence of mathematical operations. It is designated certain signs(multiplication, division, subtraction or addition). The sequence of performing mathematical actions, if necessary, is corrected by brackets. To do math is to find.

What is not an expression

Not every mathematical notation can be attributed to the number of expressions.

Equalities are not expressions. Whether or not mathematical operations are present in equality does not matter. For example, a = 5 is an equality, not an expression, but 8 + 6 * 2 = 20 also cannot be considered an expression, although it contains multiplication. This example also belongs to the category of equalities.

The concepts of expression and equality are not mutually exclusive; the former are part of the latter. The equal sign connects two expressions:
5+7=24:2

You can simplify this equality:
5+7=12

An expression always assumes that the mathematical operations presented in it can be performed. 9 +: - 7 is not an expression, although there are signs of mathematical actions, because these actions cannot be performed.

There are also some mathematical ones that are formally expressions, but do not make sense. An example of such an expression:
46:(5-2-3)

The number 46 must be divided by the result of the actions in parentheses, and it is equal to zero. You cannot divide by zero, the action is considered forbidden.

Numerical and Algebraic Expressions

There are two kinds of mathematical expressions.

If an expression contains only numbers and signs of mathematical operations, the expression is called numeric. If in the expression, along with numbers, there are variables denoted by letters, or there are no numbers at all, the expression consists only of variables and signs of mathematical operations, it is called algebraic.

The fundamental difference numerical value from algebraic is that a numerical expression has only one meaning. For example, the value of a numeric expression 56-2 * 3 will always be 50, nothing can be changed. An algebraic expression can have many meanings, because you can substitute any number instead. So, if in the expression b – 7 instead of b substitute 9, the value of the expression will be 2, and if 200 - it will be 193.

Sources:

• Numerical and Algebraic Expressions

A notation that consists of numbers, signs and brackets, and also makes sense, called a numeric expression.

For example, the following entries:

• (100-32)/17,
• 2*4+7,
• 4*0.7 -3/5,
• 1/3 +5/7

will be numeric expressions. It should be understood that one number will also be a numerical expression. In our example, this number is 13.

And, for example, the following entries

• 100 - *9,
• /32)343

will not be numerical expressions, since they are meaningless and are just a collection of numbers and signs.

Numeric expression value

Since the signs of arithmetic operations are included as signs in numerical expressions, we can calculate the value of a numerical expression. To do this, you need to follow these steps.

For example,

(100-32) / 17 = 4, that is, for the expression (100-32) / 17 the value of this numerical expression will be the number 4.

2 * 4 + 7 = 15, the number 15 will be the value of the numeric expression 2 * 4 + 7.

Often, for brevity, they do not write the full value of a numeric expression, but simply write "the value of the expression", while omitting the word "numeric".

Numeric Equality

If two numeric expressions are written with an equal sign, then these expressions form a numeric equality. For example, the expression 2 * 4 + 7 = 15 is a numeric equality.

As noted above, parentheses can be used in numeric expressions. As you already know, parentheses affect the order of actions.

In general, all actions are divided into several stages.

• First step actions: addition and subtraction.
• Actions of the second stage: multiplication and division.
• The actions of the third stage are squaring and cubing.

Rules for evaluating the values ​​of numeric expressions

When calculating the values ​​of numerical expressions, the following rules should be followed.

• 1. If the expression does not have parentheses, then it is necessary to perform actions starting from the highest levels: the third step, the second step and the first step. If there are several actions of one step, then they are performed in the order in which they are written, that is, from left to right.
• 2. If the expression contains brackets, then the actions in the brackets are performed first, and only then all the steel actions in the usual order. When performing actions in brackets, if there are several of them, you should use the order described in paragraph 1.
• 3. If the expression is a fraction, then the values ​​in the numerator and denominator are calculated first, and then the numerator is divided by the denominator.
• 4. If the expression contains nested parentheses, then the actions should be performed from the inner parentheses.

So, if a numerical expression is made up 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 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 14-215: 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 equal to 6.

Answer:

14−2 15: 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 example, .

In numerical expressions, the roots should be perceived as some numbers, and it is advisable to immediately replace the roots with their values, and then find the value of the resulting expression without roots, performing actions in the accepted sequence.

Example.

Find the meaning of the expression with roots.

Solution.

First, we find the value of the root ... For 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 calculate 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 the power 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 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, there are 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 are 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. Thus, . 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 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 of the initially specified 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 finding 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 it in advance using. 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, rewrite the base of the logarithm in the denominator as common 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 this case, we need the formula for the cosine of a double angle and the formula for the cosine of the sum:

The performed transformations helped us find the meaning of the expression.

Answer:

.

General case

V general case a numerical expression can contain roots, degrees, 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 fractions complex kind, 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:.

Thus, .

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

Calculating the values ​​of numeric expressions requires consistency and care. Yes, you must adhere to the sequence of actions written in the previous paragraphs, but you do not 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 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 execution of 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 subtraction property equal numbers: if you subtract an equal number from a number, then the result will be zero. This property can be considered more broadly: the difference 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 no less 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 value of a 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, note that sometimes the transformation literal expressions and variable expressions 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 1 for all positive values x, so the range of admissible values ​​of the variable x in the original expression is the set of positive numbers, and equality takes place on this range.

Bibliography.

• Maths: 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.
• Maths. 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.

Now that we have learned how to add and multiply individual fractions, we can consider more complex structures... For example, what if the same problem contains addition, subtraction, and multiplication of fractions?

First of all, you need to translate all fractions into incorrect ones. Then we sequentially perform the required actions - in the same order as for ordinary numbers. Namely:

1. The exponentiation is performed first - get rid of all expressions containing indicators;
2. Then - division and multiplication;
3. The last step is addition and subtraction.

Of course, if parentheses are present in the expression, the order of actions changes - everything inside the parentheses must be counted first. And remember about incorrect fractions: you need to select the whole part only when all other actions have already been completed.

Let's translate all fractions from the first expression into incorrect ones, and then perform the following actions:

Now let's find the value of the second expression. Here fractions with whole part no, but there are parentheses, so we do addition first, 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. Taking into account that 9 = 3 3, we have:

Take a look at 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 in a different way. If we recall the definition of the degree, the problem will be reduced to ordinary multiplication fractions:

Multi-storey 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 numeric fraction given in the very first lesson.

But what if a more complex object is placed in the numerator or denominator? For example, another number fraction? Such constructions occur quite often, especially when working with long expressions. Here are a couple of examples:

There is only one rule for working with multi-storey fractions: you must immediately get rid of them. Removing "extra" floors is quite easy if you remember that the fractional bar means the standard division operation. Therefore, any fraction can be rewritten as follows:

Using this fact and observing the order of actions, we can easily reduce any multi-level fraction to a regular one. Take a look at examples:

Task. Convert multi-storey fractions to regular 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.

The specifics of working with multi-level fractions

There is one subtlety in multi-storey fractions that must always be remembered, otherwise you can get the wrong answer, even if all the calculations were correct. Take a look:

1. The numerator contains a single number 7, and the denominator contains the fraction 12/5;
2. The numerator contains the fraction 7/12, and the denominator is the single number 5.

So, for one recording, we got two completely different interpretations. If you count, the answers will also be different:

To always read the entry unambiguously, use a simple rule: the separating line of the main fraction must be longer than the nested line. It is desirable - several times.

If you follow this rule, then the above fractions should be written as follows:

Yes, it might be ugly and take up too much space. But you will count correctly. Finally, a couple of examples where multi-storey fractions really occur:

Task. Find the values ​​of the expressions:

So, we are working with the first example. Let's convert all fractions to irregular ones, and then perform addition and division operations:

Let's do the same with the second example. Let's translate all fractions into irregular ones and perform the required operations. In order not to tire the reader, I will omit some of the obvious calculations. We have:

Due to the fact that there are sums in the numerator and denominator of the main fractions, the rule for writing multi-storey fractions is observed automatically. Also, in the last example, we intentionally left 46/1 in fractional form to do division.

Also note that in both examples, the fractional bar actually replaces the parentheses: first of all, we found the sum, and only then - the quotient.

Some might say that the transition to improper fractions in the second example was clearly redundant. Perhaps it is so. But by this we insure ourselves against mistakes, because next time the example may turn out to be much more complicated. Choose for yourself which is more important: speed or reliability.