This article discusses how to find the values \u200b\u200bof mathematical expressions. Let's start with simple numeric expressions and then we will consider cases as they increase their complexity. At the end we give an expression containing letter notation, brackets, roots, special mathematical signs, degrees, functions, etc. All the theory, according to tradition, supply abundant and detailed examples.

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How to find a value of a numerical expression?

Numerical expressions, among other things, help describe the condition of the problem with the mathematical language. At all mathematical expressions It can be both very simple, consisting of a pair of numbers and arithmetic signs, and very complex, containing functions, degrees, roots, brackets, etc. As part of the task, it is often necessary to find a value of a particular expression. About how to do it, and will be discussed below.

Simplest cases

These cases when the expression does not contain anything, except for numbers and arithmetic action. To successfully find the values \u200b\u200bof such expressions, you need knowledge of the procedure for performing arithmetic action without brackets, as well as the ability to perform actions with different numbers.

If there are only numbers and arithmetic signs "+", "·", "-", "÷", "," - "," ÷ ", then the actions are performed from left to right in next order: First, multiplication and division, then addition and subtraction. We give examples.

Example 1. Numerical expression value

Let it be necessary to find the values \u200b\u200bof expression 14 - 2 × 15 ÷ 6 - 3.

Perform first multiplication and division. We get:

14 - 2 · 15 ÷ 6 - 3 \u003d 14 - 30 ÷ 6 - 3 \u003d 14 - 5 - 3.

Now we carry out the subtraction and get the final result:

14 - 5 - 3 = 9 - 3 = 6 .

Example 2. Numerical expression value

Calculate: 0, 5 - 2 · - 7 + 2 3 ÷ 2 3 4 · 11 12.

First, we carry out the conversion of fractions, division and multiplication:

0, 5 - 2 · - 7 + 2 3 ÷ 2 3 4 · 11 12 \u003d 1 2 - (- 14) + 2 3 ÷ 11 4 · 11 12

1 2 - (- 14) + 2 3 ÷ 11 4 · 11 12 \u003d 1 2 - (- 14) + 2 3 · 4 11 · 11 12 \u003d 1 2 - (- 14) + 2 9.

Now we will deal with addiction and subtraction. Grouting the fraction and give them to a common denominator:

1 2 - (- 14) + 2 9 = 1 2 + 14 + 2 9 = 14 + 13 18 = 14 13 18 .

The desired value is found.

Expressions with brackets

If the expression contains brackets, they determine the procedure in this expression. First, actions are performed in brackets, and then all the others. Show it on the example.

Example 3. Numerical expression value

Find the expression value of 0, 5 · (0, 76 - 0, 06).

In the expression there are brackets, so first perform the operation of subtraction in brackets, and only later - multiplication.

0, 5 · (0, 76 - 0, 06) \u003d 0, 5 · 0, 7 \u003d 0, 35.

The value of expressions containing brackets in brackets is located on the same principle.

Example 4. Numerical expression value

We calculate the value of 1 + 2 · 1 + 2 · 1 + 2 · 1 - 1 4.

Execute actions will be starting with the most internal brackets, moving to external.

1 + 2 · 1 + 2 · 1 + 2 · 1 - 1 4 \u003d 1 + 2 · 1 + 2 · 1 + 2 · 3 4

1 + 2 · 1 + 2 · 1 + 2 · 3 4 \u003d 1 + 2 · 1 + 2 · 2, 5 \u003d 1 + 2 · 6 \u003d 13.

In finding values \u200b\u200bof expressions with brackets, the main thing is to follow the sequence of actions.

Expressions with roots

Mathematical expressions whose values \u200b\u200bwe need to find may contain root signs. Moreover, the expression itself can be under the sign of the root. How to be in this case? First you need to find the value of the expression under the root, and then remove the root from the number obtained as a result. If possible from the roots in numerical expressions, it is better to get rid of, replacing from numerical values.

Example 5. Numerical expression value

Calculate the value of the expression with the roots - 2 · 3 - 1 + 60 ÷ 4 3 + 3 · 2, 2 + 0, 1 · 0, 5.

First calculate the feeding expressions.

2 · 3 - 1 + 60 ÷ 4 3 \u003d - 6 - 1 + 15 3 \u003d 8 3 \u003d 2

2, 2 + 0, 1 · 0, 5 \u003d 2, 2 + 0, 05 \u003d 2, 25 \u003d 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 \u003d 2 + 3 · 1, 5 \u003d 6, 5

Often finding the value of expression with roots often need to first carry out the transformation of the original expression. Let's explain it to another example.

Example 6. Numerical expression value

How many will be 3 + 1 3 - 1 - 1

As you can see, we do not have the opportunity to replace the root with an accurate value that complicates the account process. However, in this case, you can apply the formula for abbreviated multiplication.

3 + 1 3 - 1 = 3 - 1 .

In this way:

3 + 1 3 - 1 - 1 = 3 - 1 - 1 = 1 .

Expressions with degrees

If there are degrees in expression, their values \u200b\u200bmust be calculated before starting all other actions. It happens that the indicator itself or the foundation of the degree is expressions. In this case, first calculate the value of these expressions, and then the value of the degree.

Example 7. The value of a numerical expression

Find the value of expression 2 3 · 4 - 10 + 16 1 - 1 2 3, 5 - 2 · 1 4.

We begin to calculate in order.

2 3 · 4 - 10 \u003d 2 12 - 10 \u003d 2 2 \u003d 4

16 · 1 - 1 2 3, 5 - 2 · 1 4 \u003d 16 * 0, 5 3 \u003d 16 · 1 8 \u003d 2.

It remains only to carry out the operation addition and find out the value of the expression:

2 3 · 4 - 10 + 16 1 - 1 2 3, 5 - 2 · 1 4 \u003d 4 + 2 \u003d 6.

It is also often advisable to simplify expression using the degree properties.

Example 8. Numerical expression value

Calculate value next expression: 2 - 2 5 · 4 5 - 1 + 3 1 3 6.

The indicators of degrees are again such that their exact numeric values \u200b\u200bwill not be able to receive. Simplify the initial expression to find its value.

2 - 2 5 · 4 5 - 1 + 3 1 3 6 \u003d 2 - 2 5 · 2 2 5 - 1 + 3 1 3 · 6

2 - 2 5 · 2 2 5 - 1 + 3 1 3 · 6 \u003d 2 - 2 5 · 2 2 · 5 - 2 + 3 2 \u003d 2 2 · 5 - 2 - 2 5 + 3 2

2 2 · 5 - 2 - 2 5 + 3 2 \u003d 2 - 2 + 3 \u003d 1 4 + 3 \u003d 3 1 4

Expressions with fractions

If the expression contains a fraction, then when calculating such an expression, all the fractions should be represented in the form of ordinary fractions and calculate their values.

If expressions are present in the numerator and denominator, the values \u200b\u200bof these expressions are calculated, and the final value of the fraction itself is written. Arithmetic actions are performed in the standard order. Consider the solution of the example.

Example 9. Numerical expression value

Find the value of the expression containing the fraction: 3, 2 2 - 3 · 7 - 2 · 3 6 ÷ 1 + 2 + 3 9 - 6 ÷ 2.

As you can see, there are three fractions in the initial expression. We calculate their values \u200b\u200bfirst.

3, 2 2 \u003d 3, 2 ÷ 2 \u003d 1, 6

7 - 2 · 3 6 \u003d 7 - 6 6 \u003d 1 6

1 + 2 + 3 9 - 6 ÷ 2 \u003d 1 + 2 + 3 9 - 3 \u003d 6 6 \u003d 1.

We rewrite our expression and calculate its value:

1, 6 - 3 · 1 6 ÷ 1 \u003d 1, 6 - 0, 5 ÷ 1 \u003d 1, 1

Often, when expressing values, it is convenient to reduce fractions. There is a checked rule: any expression before finding its value is best to simplify the maximum, reducing all the calculations to the simplest cases.

Example 10. Numerical expression value

We calculate the expression 2 5 - 1 - 2 5 - 7 4 - 3.

We cannot raise the root of five, however, we can simplify the initial expression by transformations.

2 5 - 1 = 2 5 + 1 5 - 1 5 + 1 = 2 5 + 1 5 - 1 = 2 5 + 2 4

The initial expression takes the form:

2 5 - 1 - 2 5 - 7 4 - 3 = 2 5 + 2 4 - 2 5 - 7 4 - 3 .

Calculate the value of this expression:

2 5 + 2 4 - 2 5 - 7 4 - 3 = 2 5 + 2 - 2 5 + 7 4 - 3 = 9 4 - 3 = - 3 4 .

Expressions with logarithms

When logarithms are present in the expression, their value, if possible, is calculated from the very beginning. For example, in the expression log 2 4 + 2 · 4 you can immediately write the value of this logarithm, and then perform all actions. We obtain: Log 2 4 + 2 · 4 \u003d 2 + 2 · 4 \u003d 2 + 8 \u003d 10.

Under the sign of the logarithm itself and in its foundation there may also be numerical expressions. In this case, the first thing is their meanings. Take the expression Log 5 - 6 ÷ 3 5 2 + 2 + 7. We have:

log 5 - 6 ÷ 3 5 2 + 2 + 7 \u003d log 3 27 + 7 \u003d 3 + 7 \u003d 10.

If it is impossible to calculate the exact value of the logarithm, the simplification of the expression helps to find its value.

Example 11. Numerical expression value

We will 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 \u003d log 2 8 \u003d 3.

By the property of logarithms:

log 6 2 + Log 6 3 \u003d log 6 (2 · 3) \u003d log 6 6 \u003d 1.

Recommending the properties of logarithms, for the last fraction in the expression we obtain:

log 5 729 log 0, 2 27 \u003d log 5 729 Log 1 5 27 \u003d log 5 729 - log 5 27 \u003d - log 27 729 \u003d - log 27 27 2 \u003d - 2.

Now you can go to the calculation of the value of the original expression.

log 2 log 2 256 + log 6 + log 6 3 + log 5 729 log 0, 2 27 \u003d 3 + 1 + - 2 \u003d 2.

Expressions with trigonometric functions

It happens that in the expression there are trigonometric functions of sinus, cosine, tangent and catangent, as well as functions, reverse them. From the value are calculated before performing all other arithmetic action. Otherwise, the expression is simplified.

Example 12. Numerical expression value

Find the value of the expression: T G 2 4 π 3 - SIN - 5 π 2 + COSπ.

First, calculate the values \u200b\u200bof the trigonometric functions included in the expression.

sin - 5 π 2 \u003d - 1

We substitute the values \u200b\u200bin the expression and calculate its value:

t G 2 4 π 3 - SIN - 5 π 2 + COSπ \u003d 3 2 - (- 1) + (- 1) \u003d 3 + 1 - 1 \u003d 3.

The expression value is found.

Often in order to find the value of expression with trigonometric functionsIt is pre-converted it. Let us explain on the example.

Example 13. Numerical expression value

It is necessary to find the value of the expression COS 2 π 8 - SIN 2 π 8 COS 5 π 36 COS π 9 - SIN 5 π 36 SIN π 9 - 1.

For conversion, we will use trigonometric formulas Cosuine dual angle and cosine amount.

cOS 2 π 8 - SIN 2 π 8 COS 5 π 36 COS π 9 - SIN 5 π 36 SIN π 9 - 1 \u003d COS 2 π 8 COS 5 π 36 + π 9 - 1 \u003d COS π 4 COS π 4 - 1 \u003d 1 - 1 \u003d 0.

General case of numerical expression

IN general Trigonometric expression may contain all the elements described above: brackets, degrees, roots, logarithms, functions. Formulate general rule finding values \u200b\u200bsuch expressions.

How to find an expression value

1. Roots, degrees, logarithms, etc. Replaced with their values.
2. Actions are performed in brackets.
3. The remaining actions are performed in order from left to right. First - multiplication and division, then addition and subtraction.

We will analyze an example.

Example 14. Numerical expression value

Calculate, what is equal to 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. We did not accidentally chose such an example, having taking into account 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 view, first separately there are the values \u200b\u200bof the numerator and denominator of the fraction, respectively. We will convert and simplify this expression.

First of all, we calculate the value of the feeding expression 2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3. To do this, you need to find the value of sinus, and expressions, which is the argument of a trigonometric function.

π 6 + 2 · 2 π 5 + 3 π 5 \u003d π 6 + 2 · 2 π + 3 π 5 \u003d π 6 + 2 · 5 π 5 \u003d π 6 + 2 π

Now you can find out the value of the sine:

sIN π 6 + 2 · 2 π 5 + 3 π 5 \u003d sin π 6 + 2 π \u003d sin π 6 \u003d 1 2.

Calculate the value of the feeding expression:

2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 \u003d 2 · 1 2 + 3 \u003d 4

2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 \u003d 4 \u003d 2.

With denominator, the fraction is increasingly:

Now we can write the value of the whole fraction:

2 · sin π 6 + 2 · 2 π 5 + 3 π 5 + 3 ln e 2 \u003d 2 2 \u003d 1.

With this in mind, we write down all the 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 \u003d 27.

In this case, we were able to calculate the exact values \u200b\u200bof the roots, logarithms, sinuses, etc. If there is no such possibility, you can try to get rid of them by mathematical transformations.

Calculation of values \u200b\u200bof expressions by rational methods

Calculate the numeric values \u200b\u200bmust be sequentially and neat. This process You can rationalize and speed up using various properties of actions with numbers. For example, it is known that the work is zero if zero is equal to at least one of the multipliers. Taking into account this property, you can immediately say that the expression 2 · 386 + 5 + 589 4 1 - sin 3 π 4 · 0 is zero. At the same time, it is not at all necessary to perform actions in the order described in the article above.

It is also convenient to use the deduction property. equal numbers. Not performing any actions, it can be ordered that the value of expression 56 + 8 - 3, 789 Ln E 2 - 56 + 8 - 3, 789 LN E 2 is also zero.

Another technique that allows you to speed up the process - the use of identical transformations such as a grouping of terms and multipliers and a common factor for brackets. A rational approach to calculating expressions with fractions is to reduce the same expressions in a numerator and denominator.

For example, we take expression 2 3 - 1 5 + 3 · 289 · 3 4 3 · 2 3 - 1 5 + 3 · 289 · 3 4. Not performing actions in brackets, but by reducing the fraction, we can say that the value of the expression is 1 3.

Finding values \u200b\u200bof expressions with variables

The value of the letter expression and expression with variables is for specific specified values \u200b\u200bof letters and variables.

Finding values \u200b\u200bof expressions with variables

To find the value of the letter expression and expressions with variables, it is necessary to substitute the specified values \u200b\u200bof the letters and variables in the original expression, after which it is possible to calculate the value of the number of the numerical expression.

Example 15. The value of the expression with variables

Calculate the value of expression 0, 5 x - y at the specified x \u003d 2, 4 and y \u003d 5.

We substitute the values \u200b\u200bof variables to the expression and calculate:

0, 5 x - y \u003d 0, 5 · 2, 4 - 5 \u003d 1, 2 - 5 \u003d - 3, 8.

Sometimes you can convert an expression so to obtain its value regardless of the values \u200b\u200bof the letters and variables. To do this, from letters and variables in expression, it is necessary to get rid of the opportunity using identical transformations, properties of arithmetic action and all possible other ways.

For example, the expression X + 3 shows, obviously, has a value of 3, and to calculate this value it is not at all necessary to know the value of the ICS variable. The value of this expression is to three for all values \u200b\u200bof the EX variable from its valid values.

One more example. The value of the expression X x is equal to one for all positive ICs.

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Numeric and algebraic expressions. Transformation of expressions.

What is an expression in mathematics? Why do I need transformations of expressions?

The question is, as they say, an interesting ... The fact is that these concepts are the basis of the whole mathematics. All mathematics consists of expressions and their transformations. Not very clear? I will explain.

Suppose you have an evil example. Very big and very complicated. Suppose you are strong in mathematics and are not afraid of anything! Can you immediately answer?

You'll have to solve This example. Consistently step by step, this example simplify. By defined rulesNaturally. Those. do transformation of expressions. How successful you will spend these transformations, so much so strong in mathematics. If you do not know how to make the right transformations, in mathematics you can not do nothing...

In order to avoid such an uncomfortable future (or present ...), it does not prevent to understand this topic.)

To start, find out what is an expression in mathematics. What numerical expression And what is algebraic expression.

What is an expression in mathematics?

Expression in mathematics - This is a very widespread concept. Almost everything, with which we are dealing in mathematics is a set of mathematical expressions. Any examples, formulas, fractions, equations and so on - it all consists of mathematical expressions.

3 + 2 is a mathematical expression. c 2 - D 2 - This is also a mathematical expression. And a healthy fraction, and even one number is all mathematical expressions. Equation, for example, this is:

5x + 2 \u003d 12

consists of two mathematical expressions connected by the sign of equality. One expression - on the left, the other is to the right.

IN general term " mathematical expression"It is used, most often not to wash. Ascend you, what is an ordinary fraction, for example? And how to answer?!

First answer: "This ... mmmm ... Such a thing ... in which ... can I write a better shot? What kind of? "

Second answer option: " Ordinary fraction - It (cheerfully and joyful!) mathematical expression which consists of a numerator and denominator! "

The second option will somehow be satisfied, right?)

Here for this purpose the phrase " mathematical expression "Very good. And right, and solid. But for practical application need to understand well specific species in mathematics .

Specific species is another matter. it other things! Each type of mathematical expressions have its own A set of rules and receptions that must be used when solving. For working with fractions - one set. To work with trigonometric expressions - the second. To work with logarithms - the third. Etc. Somewhere these rules coincide, somewhere - they differ sharply. But do not be afraid of these terrible words. Logarithms, trigonometry and other mysterious things we will explore in the relevant sections.

Here we will master (or - repeat someone as ...) two main types of mathematical expressions. Numerical expressions and algebraic expressions.

Numerical expressions.

What numerical expression? This is a very simple concept. The name itself hints that this is an expression with numbers. That is how it is. A mathematical expression composed of amongms, brackets and marks of arithmetic action is called a numerical expression.

7-3 - numerical expression.

(8 + 3.2) · 5.4 - also numerical expression.

And this monster:

also numerical expression, yes ...

The usual number, fraction, any example of calculating without ICs and other letters is all numerical expressions.

Chief sign numeric expressions - in it no letters. No. Only numbers and mathematical icons (if necessary). Everything is simple, right?

And what can be done with numerical expressions? Numerical expressions, as a rule, can be considered. To do this, it happens, it happens, to disclose brackets, change signs, to cut, change the terms of the lines - i.e. do converting expressions. But about it is slightly lower.

Here we will deal with such a funny case when with a numerical expression do not do anything.Well, quite nothing! This pleasant operation is nothing to do) - Perfect when the expression it does not make sense.

When the numeric expression does not make sense?

Understandable, if we see some kind of abracadabra, like

then do nothing and we will not. Since it is not clear what to do about it. Some kind of nonsense. Is that, to calculate the number of pluses ...

But there are externally well-decent expressions. For example, this:

(2 + 3): (16 - 2 · 8)

However, this expression too it does not make sense! For the simple reason that in the second brackets - if you consider - it turns out zero. And on zero it is impossible to share! This is a forbidden operation in mathematics. Therefore, it is not necessary to do anything with this expression. With any task with such an expression, the answer will always be one: "The expression does not make sense!"

To give such an answer, I had to, of course, consider that there would be in brackets. And sometimes there are such in brackets, ... well, nothing can be done here.

Forbidden operations in mathematics are not so much. In this topic - only one. Division by zero. Additional bans arising in roots and logarithms are discussed in relevant topics.

So, the idea of \u200b\u200bwhat is numerical expression - received. Concept numerical expression does not make sense - realized. Going on.

Algebraic expressions.

If letters appear in numerical expression - this expression becomes ... the expression becomes ... Yes! It becomes algebraic expression. For example:

5a 2; 3x-2y; 3 (Z-2); 3.4m / n; x 2 + 4x-4; (A + B) 2; ...

More such expressions called lettering expressions. Or expressions with variables. This is practically the same thing. Expression 5A + S., For example, both alphabetic, algebraic, and expression with variables.

Concept algebraic expression - wider than numerical. It includes And all numerical expressions. Those. Numerical expression is also an algebraic expression, only without letters. Every selence - fish, but not every fish - the selline ...)

Why literal - understandably. Well, since the letters are ... phrase expression with variables Also not much puzzles. If you understand that numbers are hidden under the letters. All numbers can be hidden under the letters ... and 5, and -18, and whatever. That is, the letter can be replace on the different numbers. Therefore, the letters are called variables.

In expression in + 5., eg, w. - variable value. Or they say simply " variable", without the word "value". Unlike the top five, which is a permanent value. Or simply - constant.

Term algebraic expression means that to work with this expression you need to use laws and rules algebra. If a arithmetic Works with specific numbers, then algebra - with all numbers. A simple example for explanation.

In arithmetic you can write down that

But if we write similar equality through algebraic expressions:

a + b \u003d b + a

we will immediately decide everything Questions. For all numbers stroke. For all endless quantity. Because under the letters but and b. Meaning everything numbers. And not only numbers, but even other mathematical expressions. That's how the algebra works.

When an algebraic expression does not make sense?

About numerical expression everything is clear. It is impossible to share on zero. And with letters, can you find out what we divide?!

Take for example here such an expression with variables:

2: (but - 5)

Does it mean? Yes, who knows him? but - Any number ...

Anyone anyone ... but there is one value butin which this expression for sure It does not make sense! And what is the number? Yes! This is 5! If the variable but Replace (say - "substitute") to the number 5, in brackets zero will turn out. Which is impossible to share. So it turns out that our expression it does not make sense, if a a \u003d 5.. But at the other values but Is there a meaning? Other numbers can be substituted?

Sure. Just in such cases say the expression

2: (but - 5)

makes sense for any values but, except a \u003d 5 .

The whole set of numbers that can substitute in a given expression called the area of \u200b\u200bpermissible values This expression.

As you can see, nothing is cunning. We look at the expression with variables, yes we understand: with what value of the variable is the forbidden operation (division on zero)?

And then we definitely look at the question of the task. What are you asking?

it does not make sense, our forbidden value and will be the answer.

If you ask, with what value variable expression has the meaning (Feel the difference!), the answer will be all other numbersexcept for forbidden.

Why do we need the meaning of expression? He is, there is no it ... What's the difference?! The fact is that this concept becomes very important in high school. Extremely important! This is the basis for such solid concepts as the area of \u200b\u200bpermissible values \u200b\u200bor the function of determining the function. Without this, you can not solve serious equations or inequalities at all. Like this.

Transformation of expressions. Identical transformations.

We got acquainted with numeric and algebraic expressions. They understood what the phrase means "the expression does not make sense." Now we must figure out what transformation of expressions. The answer is simple, to disgrace.) This is any action with the expression. And that's it. You did these transformations from the first class.

Take a steep numerical expression 3 + 5. How can it be converted? Yes, very simple! Calculate

This calculation is the transformation of the expression. You can write down the same expression differently:

Here we did not count anything at all. Just wrote an expression in another form. This will also be the transformation of the expression. You can write like this:

And this is also - the transformation of the expression. Such transformations can be posed how much you want.

Anyone action above the expression any Recording it in another form is called an expression conversion. And all things. Everything is very simple. But there is one very important rule. So important that it can be boldable chief rule All mathematics. Violation of this rule inevitably leads to errors. Invibly?)

Suppose we transformed our expression as it fell, like this:

Transformation? Sure. We recorded the expression in another form, what's wrong here?

Everything is wrong.) The fact is that transformations "As hit" Mathematics are not interested at all.) All mathematics are built on transformations in which it changes appearance, but the essence of the expression does not change. Three plus five can be written in any way, but it should be eight.

Transformations not changing the essence of the expression called identical.

Exactly identical transformations and allow us step by step, turn complex example In a simple expression, keeping essence of the example. If we make a mistake in the chain of transformations, we do not identical conversion, then we will decide already other example. With other answers that are not related to the right.)

Here it is the main rule of solving any tasks: compliance with the identity of transformations.

An example with numeric expression 3 + 5 I brought for clarity. In algebraic expressions, identical transformations are given by formulas and rules. Say, there is a formula in algebra:

a (B + C) \u003d AB + AC

So we can instead of expression in any example a (B + C) Boldly write an expression aB + AC. And vice versa. it identical conversion. Mathematics gives us a choice of these two expressions. And which of them write - from specific example Depends on

Another example. One of the most important and necessary transformations is the main property of the fraction. You can see more details, and here you just remember the rule: if the numerator and denominator of the fraci multiply (divided) per and the same number, or unequal zero expression, the fraction will not change. Here is an example of identical transformations on this property:

As you probably guessed, this chain can be continued to infinity ...) a very important property. It is it that allows you to turn any monsters-examples in white and fluffy.)

The formulas specifying identical transformations - a lot. But the most important is quite a reasonable amount. One of the basic transformations is the decomposition of multipliers. It is used in the whole mathematics - from elementary to the highest. From him and start. In the next lesson.)

If you like this site ...

By the way, I have another couple of interesting sites for you.)

It can be accessed in solving examples and find out your level. Testing with instant check. Learn - with interest!)

You can get acquainted with features and derivatives.

As a rule, children begin to study algebra already in junior classes. After mastering the basic principles of working with numbers, they solve examples with one or more unknown variables. Find the value of the expression of this plan can be quite difficult, but if you simplify it using the knowledge of elementary school, everything will be easily and quickly.

What is the meaning of expression

Numerical expression called algebraic recordingconsisting of numbers, brackets and signs in the event that it makes sense.

In other words, if you can find the value of the expression, it means that the record is not deprived of meaning, and vice versa.

Examples of the following records are correct numerical structures:

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

A separate number will also be a numerical expression as the number 18 of the above example.
Examples of incorrect numerical structures that do not make sense:

• *7-25);
• 16/0-;
• (*-5;

Incorrect numeric examples are only a set of mathematical signs and have no meaning.

How to find the value of expression

Since arithmetic signs are present in such examples, it can be concluded that they allow arithmetic calculations. To calculate signs or, speaking, to find the value of the expression, you must perform the corresponding arithmetic manipulations.

As an example, you can consider the following structure: (120-30) / 3 \u003d 30. The number 30 will be the value of the numerical expression (120-30) / 3.

Instruction:

The concept of numerical equality

Numeric equality is called the situation when two parts of the example are divided by the sign "\u003d". That is, one part is completely equal (identical) another, even if the symbols and numbers are displayed as other combinations.
For example, any design of type 2 + 2 \u003d 4 can be called a numerical equality, since even changing parts by places, the meaning will not change: 4 \u003d 2 + 2. The same applies to more complex structures, including brackets, division, multiplication, action with fractions and so on.

How to find the value of expression correctly

To correctly find the value of the expression, you must perform calculations according to a specific procedure. This procedure is also taught in the lessons of mathematics, and later - in the class of algebra in primary school. It is also known as the stage of arithmetic action.

Arithmetic steps:

1. The first step is the addition and subtraction of numbers.
2. The second stage is a division and multiplication.
3. The third stage - the numbers are erected into a square or cube.

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

1. Perform actions starting with the third stage ending first if there are no brackets in the example. That is, first build a square or cube, then divide or multiply and only then fold and subtract.
2. In designs with brackets first, perform actions in brackets, and then follow the procedure described above. If there are several brackets, also use the procedure from the first item.
3. In the examples in the form of a fraction first, find out the result in the numerator, then in the denominator, after which the first divide on the second.

It is not difficult to find an expression value if you assimilate the elementary knowledge of the initial courses of algebra and mathematics. Guided by the information described above, you can solve any task, even increased complexity.

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Transformation of expressions. Detailed theory (2019)

Transformation of expressions

Often we hear this unpleasant phrase: "Simplify the expression". Usually, in addition, we have some kind of scary of this type:

"Yes, much easier" - we say, but this answer usually does not roll.

Now I will teach you not to be afraid of any such tasks. Moreover, at the end of the lesson you will simplify this example before (just!) Ordinary number (yes, to hell with these letters).

But before proceeding to this lesson, you need to be able to handle fractions and lay the polynomials to multipliers. Therefore, first, if you did not do this before, necessarily the topic "" and "".

Read? If so, now you are ready.

Basic simplification operations

Now we will analyze the main techniques that are used in simplifying expressions.

The easiest of them is

1. Bringing similar

What are the like? You passed it in grade 7, as soon as the letters appeared in mathematics instead of numbers. Similar are the components (single) with the same alphabetic part. For example, in the amount of such components - this is.

Remembered?

Certain similar things - it means to fold several similar terms with each other and get one term.

But how do we fold with each other letters? - You ask you.

It is very easy to understand if you imagine that letters are some items. For example, the letter is a chair. Then what is the expression? Two stools plus three stools, how much will it be? That's right, chairs :.

And now try such an expression :.

In order not to get confused, let the different letters indicate different items. For example, it is (as usual) a chair, and is the table. Then:

Chair table chairs chairs at stools chairs

Numbers for which letters multiply in such terms are called coefficients. For example, in single-wing coefficient is equal. And in it is equal.

So, the rule of bringing similar:

Examples:

Give similar:

Answers:

2. (And similar, since, therefore these terms have the same letter part).

2. Decomposition of multipliers

This is usually the most important part in simplifying expressions. After you led the like, most often the resulting expression should be decomposed on multipliers, that is, to imagine in the form of a work. This is especially important in the fraud: after all, so that you can cut the fraction, the numerator and the denominator must be represented as a work.

In detail, ways to decompose expressions on multipliers, you passed in the topic "", so here you can only remember the learned. To do this, solve several examples (you need to decompose on multipliers):

Solutions:

3. Reducing the fraction.

Well, what can be more pleasant than to cross part of the numerator and denominator, and throw them away from your life?

This is all the charm of reduction.

Everything is simple:

If the numerator and denominator contain the same multipliers, they can be cut, that is, removed from the fraction.

This rule follows from the main property of the fraci:

That is, the essence of the reduction operation is that the numerator and denominator of the fractiona divide on the same number (or on the same expression).

To shorten the fraction, you need:

1) Numerator and denominator decompose on multipliers

2) if there is a numerator and denominator common multipliersThey can be deleted.

Principle, I think, is clear?

I want to pay attention to one typical mistake With a reduction. Although this topic is simple, but very many do everything wrong, not understanding that cut - it means divide Numerator and denominator per and the same number.

No abbreviations, if in a numerator or denominator amount.

For example: it is necessary to simplify.

Some do this: what is absolutely wrong.

Another example: cut.

"The smartest" will do this:.

Tell me what is wrong here? It would seem: - This is a multiplier, it means you can cut.

But no: - This is the multiplier of only one term in the numerator, but the numeral itself is not laid out on the multipliers.

Here is another example :.

This expression is decomposed on multipliers, it means that you can cut, that is, to divide the numerator and denominator on, and then on:

You can immediately share on:

To prevent such mistakes, remember easy wayHow to determine whether the expression on multipliers is declined:

The arithmetic action that is performed by the last when calculating the values \u200b\u200bof the expression is the "main". That is, if you substitute any (any) numbers instead of letters, and you will try to calculate the value of the expression, if the last action is multiplication - it means that we have a work (expression is decomposed on multipliers). If the latter action is addition or subtraction, it means that the expression is not decomposed on the factors (and therefore cannot be reduced).

For consolidation, we decide on your own several examples:

Answers:

1. I hope you have not rushed immediately reduce and? Not enough "cut" such such such:

The first action should be a decomposition of multipliers:

4. Addition and subtraction of fractions. Bringing fractions to a common denominator.

Addition and subtraction of ordinary fractions - the operation is well familiar: We are looking for a common denominator, we are dominant every fraction on the missing multiplier and fold / deduct the numerals. Let's remember:

Answers:

1. The denominators are mutually simple, that is, they do not have common multipliers. Consequently, the NOC of these numbers is equal to their work. This will be a common denominator:

2. Here, the overall denominator is:

3. Here is the first thing mixed fractions We turn into incorrect, and then - by the usual scheme:

It is quite another thing if the fractions contain letters, for example:

Let's start with simple:

a) denominators do not contain letters

Here is all the same as with conventional numerical fractions: we find a common denominator, we are dominant every fraction on the missing multiplier and fold / deduct the numerals:

now in the numerator you can give similar, if any, and lay out on multipliers:

Try it yourself:

b) denominators contain letters

Let's remember the principle of finding a common denominator without letters:

· First of all, we define general factors;

· Then we write out all the general factors one time;

· And they are dominant to all other multipliers, not common.

To determine the general multipliers of the denominators, first lay them off on simple factors:

We emphasize general factors:

Now we will write down the general factors for one time and add all the options (not underlined) multipliers to them:

This is a common denominator.

Let's go back to the letters. Dannels are given by exactly the same scheme:

· Decide denominators for multipliers;

· Determine the general (identical) multipliers;

· We write all the general factors one time;

· We are dominant to all other multipliers, not common.

So, in order:

1) expand the denominators for multipliers:

2) Determine the general (identical) multipliers:

3) We write out all the general factors one time and the dominant of them on all the others (inextricted) multipliers:

So, the general denominator is here. The first fraction must be multiplying on, the second - on:

By the way, there is one trick:

For example: .

We see the same multipliers in the denominators, just all with different indicators. In the overall denominator will go:

in degree

in degree

in degree

to degree.

Complicate Task:

How to make the same denominator?

Let's remember the main property of the fraci:

Nowhere is not said that the fraction can be subtracted from the numerator and denominator) (or add) the same number. Because it is incorrect!

Clean yourself: take any fraction, for example, and add to the numerator and denominator some number, for example,. What did you say?

So, the next unshakable rule:

When you bring a fraction to a common denominator, use only multiplication operation!

But what do you need to multiply to get?

Here is on and the dominat. And the Domanki on:

Expressions that cannot be decomposed on multiplies will be called "elementary multipliers." For example, it is an elementary multiplier. - also. But - no: it is decomposed on multipliers.

What do you say about the expression? It is elementary?

No, because it can be decomposed on multipliers:

(On the decomposition of multipliers, you already read in the topic "").

So, the elementary multipliers to which you decline the expression with letters is an analogue of simple multipliers to which you spread numbers. And we will act with them in the same way.

We see that in both denominators there is a multiplier. He will go to a common denominator to a degree (remember why?).

The multiplier is elementary, and they don't have a general one, which means the first fraction on it will have to simply draw:

Another example:

Decision:

Expires than in a panic multiply these denominators, you need to think about how to decompose them for multipliers? Both of them represent:

Excellent! Then:

Another example:

Decision:

As usual, decompose the denominators for multipliers. In the first denominator, we just endure behind the brackets; In the second - the difference of squares:

It would seem that there are no general factors. But if you look at, then they are similar ... And the truth:

So write:

That is, it turned out like this: inside the bracket, we changed the places in places, and at the same time the sign was changed before the opposite. Take note, so it will have to do often.

Now we give a common denominator:

Help? Now check.

Tasks for self solutions:

Answers:

Here it is necessary to remember another one - the difference of cubes:

Pay attention that in the denominator the second fraction is not the formula "Square amount"! Square amount would look like this:.

And - this is the so-called incomplete square of the amount: the second term in it is the work of the first and last, and not doubled their work. The incomplete square of the amount is one of the multipliers in the decomposition of the difference of cubes:

What to do if fractions are already three pieces?

And the same thing! First of all, we do so that the maximum number of multipliers in the denominators was the same:

Pay attention: if you change the signs inside one bracket, the sign before the fraction is changing to the opposite. When we change the signs in the second bracket, the sign before the fraction changes again to the opposite. As a result, he (the sign before the fraction) has not changed.

In the overall denominator, the first denominator is discharged, and then add all the factors that are not written, from the second, and then from the third (and so on, if the frains are more). That is, it turns out like this:

Hmm ... with fractions, it is clear what to do. But how to be with a twos?

Everything is simple: you know how to put a fraction? So, you need to do so that the twice becomes a fraction! We remember: the fraction is a division operation (the numerator shares the denominator if you suddenly forgot). And there is nothing easier than split the number on. At the same time, the number itself will not change, but will turn into a fraction:

Exactly what is needed!

5. Multiplication and division of fractions.

Well, the most difficult now behind. And we have the simplest, but the most important thing is:

Procedure

What is the procedure for counting a numerical expression? Remember, considering the importance of such an expression:

Calculated?

Must happen.

So, I remind.

The first thing is calculated degree.

The second is multiplication and division. If multiplications and divisions are simultaneously several, you can do them in any order.

And finally, we perform addition and subtraction. Again, in any order.

But: the expression in brackets is calculated out of turn!

If several brackets are multiplied or shared on each other, we calculate the expression in each of the brackets first, and then multiply or delivered them.

And if there are still some brackets inside the brackets? Well, let's think: some expression is written inside the brackets. And when calculating the expression, first of all, you need to do what? That's right, calculate brackets. Well, so figured out: first we calculate the internal brackets, then everything else.

So, the procedure for the expression is higher than this (the current values \u200b\u200bare allocated red, that is, the action that I perform right now):

Well, it's simple.

But this is not the same as the expression with letters?

No, it's the same! Only instead of arithmetic actions should be made algebraic, that is, the actions described in previous section: bringing similar, Adjusting fractions, cutting fractions, and so on. The only difference will be the action of the decomposition of polynomials on multipliers (we often apply it when working with fractions). Most often, for decomposition on multipliers, I need to apply or simply take out a common factor for brackets.

Usually our goal is to submit an expression in the form of a work or private.

For example:

We simplify expression.

1) First we simplify the expression in brackets. There we have a difference fraction, and our goal is to present it as a work or private. So, we give a fraction for a common denominator and fold:

More this expression is easy to simplify, all the factors here are elementary (you still remember what it means?).

2) We get:

Multiplication of fractions: what could be easier.

3) Now you can reduce:

That's it. Nothing difficult, right?

Another example:

Simplify expression.

First try to solve myself, and only then see the decision.

First, we define the procedure for action. First, we will perform the addition of fractions in brackets, it turns out instead of two fractions one. Then we will perform dividing fractions. Well, the result will lay down with the last fraction. Schematically number actions:

Now I'll show the news process, tapping the current action in red:

Finally, you will give you two useful advice:

1. If there are similar, they must be brought immediately. In whatever time, we have similar similar, it is advisable to bring them immediately.

2. The same applies to the reduction of fractions: as soon as the ability to reduce, it needs to be used. The exception is the fractions that you fold or deduct: if they have the same denominators now, then the abbreviation must be left for later.

Here are your tasks for self solutions:

And promised at the very beginning:

Solutions (brief):

If you coped at least with the first three examples, then you, consider, mastered.

Now forward to learning!

Transformation of expressions. Summary and basic formulas

Basic simplification operations:

• Bringing similar: To fold (lead) similar components, it is necessary to fold their coefficients and attribute the letter part.
• Factorization:taking a common factor for brackets, application, etc.
• Reduction of fractions: The numerator and denominator of the fraction can be multiplied or divided into one and the same non-zero number, from which the fraction is not changed.
  1) Numerator and denominator decompose on multipliers
  2) If there are general multipliers in a numerator and denominator, they can be deleted.

  IMPORTANT: Only multipliers can be cut!

• Addition and subtraction of fractions:
  ;
• Multiplication and division of fractions:
  ;

Now, when we learned to fold and multiply individual fractions, you can consider more complex structures. For example, what if the task is also being addicted and subtracting, and multiplying fractions?

First of all, it is necessary to translate all the fractions into the wrong. Then we consistently perform the required actions - in the same manner as for conventional numbers. Namely:

1. First, it is raised to a degree - get rid of all expressions containing indicators;
2. Then - division and multiplication;
3. The last step is made addition and subtraction.

Of course, if there are brackets in the expression, the procedure is changed - everything that stands inside the brackets should be considered first. And remember the wrong fractions: it is necessary to allocate the whole part when all other actions have already been fulfilled.

We translate all the fractions from the first expression to the wrong, and then perform actions:

Now find the value of the second expression. Here frains S. whole part No, but there are brackets, so first perform addition, and only then - division. Note that 14 \u003d 7 · 2. Then:

Finally, we consider the third example. There are brackets and a degree - they are better considered separately. Given that 9 \u003d 3 · 3, we have:

Pay attention to the last example. To build a fraction in the extent, it is necessary to separate the numerator separately into this degree, and separately the denominator.

You can solve differently. If you recall the degree of degree, the task will be reduced to ordinary multiplication frains:

Multi-storey fractions

So far, we have only considered the "clean" fractions when the numerator and the denominator are ordinary numbers. This fully corresponds to the determination of the numerical fraction given in the very first lesson.

But what if in a numerator or denominator to place a more complex object? For example, another numeric fraction? Such structures arise quite often, especially when working with long expressions. Here is a couple of examples:

The rule of work with multi-storey fractions is only one thing: it is necessary to get rid of them immediately. Remove the "extra" floors is quite simple if you remember that the fractional feature means a standard division operation. Therefore, any fraction can be rewritten as follows:

Using this fact and observing the procedure, we will easily reduce any multi-storey fraction to normal. Take a look at the examples:

A task. Translate multi-storey fractions to normal:

In each case, rewrite the bulk fraction, replacing the separation feature of the division. We also remember that any integer represents in the form of a fraction with denominator 1. Those. 12 \u003d 12/1; 3 \u003d 3/1. We get:

In the last example, the final multiplication of the fraction was reduced.

Specificity of working with multi-storey fractions

In multi-storey fractions there is one subtlety that you always need to remember, otherwise you can get an incorrect answer, even if all calculations were correct. Take a look:

1. The numener is a separate number 7, and in the denominator - the shot 12/5;
2. In the numerator there is a fraction 7/12, and in the denominator - a separate number 5.

So, for one record, two completely different interpretations received. If calculating, the answers will also be different:

To record always read definitely, use a simple rule: the shared line of the main fraction should be longer than the trait nested. Preferably - several times.

If you follow this rule, the above-handed fractions should be recorded like this:

Yes, perhaps it is ugly and takes too much space. But you will consider correctly. Finally, a pair of examples where multi-storey fractions really occur:

A task. Find the values \u200b\u200bof expressions:

So, we work with the first example. We translate all the fractions into the wrong, and then perform the operations of addition and division:

Similarly, proceed with the second example. We translate all the fractions into the wrong and execute the required operations. In order not to tire the reader, I will give some obvious calculations. We have:

Due to the fact that in the numerator and denominator of the main fractions there are amounts, the recording rule of multi-storey frains is automatically respected. In addition, in the latter example, we intentionally left the number 46/1 in the form of a fraction to fulfill the division.

I also note that in both examples, a fractional feature actually replaces brackets: the first thing we found the amount, and only then are private.

Someone will say that the transition to the wrong fractions in the second example was clearly excessive. Maybe it is. But by this we insure ourselves from errors, because the next time an example may be much more complicated. Choose yourself, more importantly: speed or reliability.