Any language can express the same information different words and turnovers. Mathematical language is no exception. But the same expression can be equivalently written in different ways. And in some situations, one of the entries is simpler. We will talk about simplifying expressions in this lesson.

People communicate on different languages. For us, an important comparison is the pair "Russian language - mathematical language". The same information can be reported in different languages. But, besides this, it can be pronounced differently in one language.

For example: “Peter is friends with Vasya”, “Vasya is friends with Petya”, “Peter and Vasya are friends”. Said differently, but one and the same. By any of these phrases, we would understand what is at stake.

Let's look at this phrase: "The boy Petya and the boy Vasya are friends." We understand what in question. However, we don't like how this phrase sounds. Can't we simplify it, say the same, but simpler? “Boy and boy” - you can say once: “Boys Petya and Vasya are friends.”

"Boys" ... Isn't it clear from their names that they are not girls. We remove the "boys": "Petya and Vasya are friends." And the word "friends" can be replaced with "friends": "Petya and Vasya are friends." As a result, the first, long, ugly phrase was replaced with an equivalent statement that is easier to say and easier to understand. We have simplified this phrase. To simplify means to say it easier, but not to lose, not to distort the meaning.

The same thing happens in mathematical language. The same thing can be said differently. What does it mean to simplify an expression? This means that for the original expression there are many equivalent expressions, that is, those that mean the same thing. And from all this multitude, we must choose the simplest, in our opinion, or the most suitable for our further purposes.

For example, consider a numeric expression. It will be equivalent to .

It will also be equivalent to the first two: .

It turns out that we have simplified our expressions and found the shortest equivalent expression.

For numeric expressions, you always need to do all the work and get the equivalent expression as a single number.

Consider an example of a literal expression . Obviously, it will be simpler.

When simplifying literal expressions, you must perform all the actions that are possible.

Is it always necessary to simplify an expression? No, sometimes an equivalent but longer notation will be more convenient for us.

Example: Subtract the number from the number.

It is possible to calculate, but if the first number were represented by its equivalent notation: , then the calculations would be instantaneous: .

That is, a simplified expression is not always beneficial for us for further calculations.

Nevertheless, very often we are faced with a task that just sounds like "simplify the expression."

Simplify the expression: .

Solution

1) Perform actions in the first and second brackets: .

2) Calculate the products: .

Obviously, the last expression has a simpler form than the initial one. We have simplified it.

In order to simplify the expression, it must be replaced with an equivalent (equal).

To determine the equivalent expression, you must:

1) perform all possible actions,

2) use the properties of addition, subtraction, multiplication and division to simplify calculations.

Properties of addition and subtraction:

1. Commutative property of addition: the sum does not change from the rearrangement of the terms.

2. Associative property of addition: in order to add a third number to the sum of two numbers, you can add the sum of the second and third numbers to the first number.

3. The property of subtracting a sum from a number: to subtract the sum from a number, you can subtract each term individually.

Properties of multiplication and division

1. The commutative property of multiplication: the product does not change from a permutation of factors.

2. Associative property: to multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second factor.

3. The distributive property of multiplication: in order to multiply a number by a sum, you need to multiply it by each term separately.

Let's see how we actually do mental calculations.

Calculate:

Solution

1) Imagine how

2) Let's represent the first factor as the sum bit terms and do the multiplication:

3) you can imagine how and perform multiplication:

4) Replace the first factor with an equivalent sum:

The distributive law can also be used in the opposite direction: .

Follow these steps:

1) 2)

Solution

1) For convenience, you can use the distribution law, just use it in the opposite direction - take the common factor out of brackets.

2) Let's take the common factor out of brackets

It is necessary to buy linoleum in the kitchen and hallway. Kitchen area - hallway -. There are three types of linoleums: for, and rubles for. How much will each of the three types of linoleum cost? (Fig. 1)

Rice. 1. Illustration for the condition of the problem

Solution

Method 1. You can separately find how much money it will take to buy linoleum in the kitchen, and then add it to the hallway and add up the resulting works.

Expressions, expression conversion

Power expressions (expressions with powers) and their transformation

In this article, we will talk about transforming expressions with powers. First, we will focus on the transformations that are performed with expressions of any kind, including power expressions, such as opening brackets, reducing similar terms. And then we will analyze the transformations inherent specifically in expressions with degrees: working with the base and exponent, using the properties of degrees, etc.

Page navigation.

What are Power Expressions?

The term "power expressions" is practically not found in school textbooks of mathematics, but it often appears in collections of tasks, especially designed to prepare for the Unified State Examination and the OGE, for example,. After analyzing tasks in which it is required to perform any actions with power expressions, it becomes clear that power expressions are understood as expressions containing degrees in their entries. Therefore, for yourself, you can take the following definition:

Definition.

Power expressions are expressions containing powers.

Let's bring examples of power expressions. Moreover, we will represent them according to how the development of views on from a degree with a natural indicator to a degree with a real indicator takes place.

As you know, first there is an acquaintance with the degree of a number with a natural exponent, at this stage the first simplest power expressions of the type 3 2 , 7 5 +1 , (2+1) 5 , (−0,1) 4 , 3 a 2 −a+a 2 , x 3−1 , (a 2) 3 etc.

A little later, the power of a number with an integer exponent is studied, which leads to the appearance of power expressions with negative integer powers, like the following: 3 −2, , a −2 +2 b −3 + c 2 .

In the senior classes, they return to the degrees again. There is introduced a degree with rational indicator, which leads to the appearance of the corresponding power expressions: , , etc. Finally, degrees with irrational exponents and expressions containing them are considered: , .

The matter is not limited to the listed power expressions: further the variable penetrates into the exponent, and there are, for example, such expressions 2 x 2 +1 or . And after getting acquainted with, expressions with powers and logarithms begin to appear, for example, x 2 lgx −5 x lgx.

So, we figured out the question of what are power expressions. Next, we will learn how to transform them.

The main types of transformations of power expressions

With power expressions, you can perform any of the basic identity transformations of expressions. For example, you can expand brackets, replace numeric expressions with their values, add like terms, and so on. Naturally, in this case it is necessary to follow the accepted procedure for performing actions. Let's give examples.

Example.

Calculate the value of the power expression 2 3 ·(4 2 −12) .

Solution.

According to the order of the actions, we first perform the actions in brackets. There, firstly, we replace the power of 4 2 with its value 16 (see if necessary), and secondly, we calculate the difference 16−12=4 . We have 2 3 (4 2 −12)=2 3 (16−12)=2 3 4.

In the resulting expression, we replace the power of 2 3 with its value 8 , after which we calculate the product 8·4=32 . This is the desired value.

So, 2 3 (4 2 −12)=2 3 (16−12)=2 3 4=8 4=32.

Answer:

2 3 (4 2 −12)=32 .

Example.

Simplify Power Expressions 3 a 4 b −7 −1+2 a 4 b −7.

Solution.

Obviously, this expression contains similar terms 3 · a 4 · b − 7 and 2 · a 4 · b − 7 , and we can reduce them: .

Answer:

3 a 4 b −7 −1+2 a 4 b −7 =5 a 4 b −7 −1.

Example.

Express an expression with powers as a product.

Solution.

To cope with the task allows the representation of the number 9 as a power of 3 2 and the subsequent use of the reduced multiplication formula, the difference of squares:

Answer:

There are also a number of identical transformations inherent in power expressions. Next, we will analyze them.

Working with base and exponent

There are degrees, in the basis and / or indicator of which are not just numbers or variables, but some expressions. As an example, let's write (2+0.3 7) 5−3.7 and (a (a+1)−a 2) 2 (x+1) .

When working with similar expressions, both the expression in the base of the degree and the expression in the exponent can be replaced identically equal expression on the ODZ of its variables. In other words, according to the rules known to us, we can separately convert the base of the degree, and separately - the indicator. It is clear that as a result of this transformation, an expression is obtained that is identically equal to the original one.

Such transformations allow us to simplify expressions with powers or achieve other goals we need. For example, in the power expression (2+0.3 7) 5−3.7 mentioned above, you can perform operations with numbers in the base and exponent, which will allow you to go to the power of 4.1 1.3. And after opening the brackets and bringing like terms in the base of the degree (a (a + 1) − a 2) 2 (x + 1) we get a power expression more simple form a 2 (x+1) .

Using Power Properties

One of the main tools for transforming expressions with powers is equalities that reflect . Let us recall the main ones. For any positive numbers a and b and arbitrary real numbers r and s have the following properties of degrees:

• a r a s =a r+s ;
• a r:a s =a r−s ;
• (a b) r = a r b r ;
• (a:b) r =a r:b r ;
• (a r) s =a r s .

Note that for natural, integer, and positive exponents, restrictions on the numbers a and b may not be so strict. For example, for natural numbers m and n the equality a m ·a n =a m+n is true not only for positive a , but also for negative ones, and for a=0 .

At school, the main attention in the transformation of power expressions is focused precisely on the ability to choose suitable property and apply it correctly. In this case, the bases of the degrees are usually positive, which allows you to use the properties of the degrees without restrictions. The same applies to the transformation of expressions containing variables in the bases of degrees - the range of acceptable values ​​​​of variables is usually such that the bases take only positive values ​​on it, which allows you to freely use the properties of degrees. In general, you need to constantly ask yourself whether it is possible to apply any property of degrees in this case, because inaccurate use of properties can lead to a narrowing of the ODZ and other troubles. These points are discussed in detail and with examples in the article transformation of expressions using the properties of degrees. Here we confine ourselves to a few simple examples.

Example.

Express the expression a 2.5 ·(a 2) −3:a −5.5 as a power with base a .

Solution.

First, we transform the second factor (a 2) −3 by the property of raising a power to a power: (a 2) −3 =a 2 (−3) =a −6. In this case, the initial power expression will take the form a 2.5 ·a −6:a −5.5 . Obviously, it remains to use the properties of multiplication and division of powers with the same base, we have
a 2.5 a -6:a -5.5 =
a 2.5−6:a−5.5 =a−3.5:a−5.5 =
a −3.5−(−5.5) =a 2 .

Answer:

a 2.5 (a 2) -3:a -5.5 \u003d a 2.

Power properties are used when transforming power expressions both from left to right and from right to left.

Example.

Find the value of the power expression.

Solution.

Equality (a·b) r =a r ·b r , applied from right to left, allows you to go from the original expression to the product of the form and further. And when multiplying powers with the same grounds indicators add up: .

It was possible to perform the transformation of the original expression in another way:

Answer:

.

Example.

Given a power expression a 1.5 −a 0.5 −6 , enter a new variable t=a 0.5 .

Solution.

The degree a 1.5 can be represented as a 0.5 3 and further on the basis of the property of the degree in the degree (a r) s =a r s applied from right to left, convert it to the form (a 0.5) 3 . In this way, a 1.5 -a 0.5 -6=(a 0.5) 3 -a 0.5 -6. Now it is easy to introduce a new variable t=a 0.5 , we get t 3 −t−6 .

Answer:

t 3 −t−6 .

Converting fractions containing powers

Power expressions can contain fractions with powers or represent such fractions. Any of the basic fraction transformations that are inherent in fractions of any kind are fully applicable to such fractions. That is, fractions that contain degrees can be reduced, reduced to a new denominator, work separately with their numerator and separately with the denominator, etc. To illustrate the above words, consider the solutions of several examples.

Example.

Simplify Power Expression .

Solution.

This power expression is a fraction. Let's work with its numerator and denominator. In the numerator, we open the brackets and simplify the expression obtained after that using the properties of powers, and in the denominator we present similar terms:

And we also change the sign of the denominator by placing a minus in front of the fraction: .

Answer:

.

Reduction of containing powers of fractions to a new denominator is carried out similarly to reduction to a new denominator rational fractions. At the same time, an additional factor is also found and the numerator and denominator of the fraction are multiplied by it. When performing this action, it is worth remembering that reduction to a new denominator can lead to a narrowing of the DPV. To prevent this from happening, it is necessary that the additional factor does not vanish for any values ​​of the variables from the ODZ variables for the original expression.

Example.

Bring the fractions to a new denominator: a) to the denominator a, b) to the denominator.

Solution.

a) In this case, it is quite easy to figure out what additional factor helps to achieve the desired result. This is a multiplier a 0.3, since a 0.7 a 0.3 = a 0.7+0.3 = a . Note that in the range of acceptable values ​​of the variable a (this is the set of all positive real numbers), the degree a 0.3 does not vanish, therefore, we have the right to multiply the numerator and denominator of the given fraction by this additional factor:

b) Looking more closely at the denominator, we find that

and multiplying this expression by will give the sum of cubes and , that is, . And this is the new denominator to which we need to bring the original fraction.

So we found an additional factor . The expression does not vanish on the range of acceptable values ​​of the variables x and y, therefore, we can multiply the numerator and denominator of the fraction by it:

Answer:

but) , b) .

There is also nothing new in the reduction of fractions containing degrees: the numerator and denominator are represented as a certain number of factors, and the same factors of the numerator and denominator are reduced.

Example.

Reduce the fraction: a) , b).

Solution.

a) First, the numerator and denominator can be reduced by the numbers 30 and 45, which equals 15. Also, obviously, you can reduce by x 0.5 +1 and by . Here's what we have:

b) In this case, the same factors in the numerator and denominator are not immediately visible. To get them, you have to perform preliminary transformations. In this case, they consist in decomposing the denominator into factors according to the difference of squares formula:

Answer:

but)

b) .

Reducing fractions to a new denominator and reducing fractions is mainly used to perform operations on fractions. Actions are performed according to known rules. When adding (subtracting) fractions, they are reduced to a common denominator, after which the numerators are added (subtracted), and the denominator remains the same. The result is a fraction whose numerator is the product of the numerators, and the denominator is the product of the denominators. Division by a fraction is multiplication by its reciprocal.

Example.

Follow the steps .

Solution.

First, we subtract the fractions in brackets. To do this, we bring them to a common denominator, which is , then subtract the numerators:

Now we multiply fractions:

Obviously, a reduction by the power x 1/2 is possible, after which we have .

You can also simplify the power expression in the denominator by using the difference of squares formula: .

Answer:

Example.

Simplify Power Expression .

Solution.

Obviously, this fraction can be reduced by (x 2.7 +1) 2, this gives the fraction . It is clear that something else needs to be done with the powers of x. To do this, we convert the resulting fraction into a product. This gives us the opportunity to use the property of dividing powers with the same bases: . And at the end of the process, we pass from the last product to the fraction.

Answer:

.

And we add that it is possible and in many cases desirable to transfer factors with negative exponents from the numerator to the denominator or from the denominator to the numerator by changing the sign of the exponent. Such transformations often simplify further actions. For example, a power expression can be replaced by .

Converting expressions with roots and powers

Often in expressions in which some transformations are required, along with degrees with fractional exponents, there are also roots. To convert such an expression to the right kind, in most cases it is enough to go only to roots or only to powers. But since it is more convenient to work with degrees, they usually move from roots to degrees. However, it is advisable to carry out such a transition when the ODZ of variables for the original expression allows you to replace the roots with degrees without the need to access the module or split the ODZ into several intervals (we discussed this in detail in the article, the transition from roots to powers and vice versa After getting acquainted with the degree with a rational exponent a degree with an irrational indicator is introduced, which makes it possible to speak of a degree with an arbitrary real indicator.At this stage, the school begins to study exponential function , which is analytically given by the degree, in the basis of which there is a number, and in the indicator - a variable. So we are faced with exponential expressions containing numbers in the base of the degree, and in the exponent - expressions with variables, and naturally the need arises to perform transformations of such expressions.

It should be said that the transformation of expressions of the indicated type usually has to be performed when solving exponential equations And exponential inequalities , and these transformations are quite simple. In the vast majority of cases, they are based on the properties of the degree and are aimed mostly at introducing a new variable in the future. The equation will allow us to demonstrate them 5 2 x+1 −3 5 x 7 x −14 7 2 x−1 =0.

First, the exponents, in whose exponents the sum of some variable (or expression with variables) and a number, is found, are replaced by products. This applies to the first and last terms of the expression on the left side:
5 2 x 5 1 −3 5 x 7 x −14 7 2 x 7 −1 =0,
5 5 2 x −3 5 x 7 x −2 7 2 x =0.

Next, both parts of the equality are divided by the expression 7 2 x , which takes only positive values ​​on the ODZ of the variable x for the original equation (this is a standard technique for solving equations of this kind, we are not talking about it now, so focus on subsequent transformations of expressions with powers ):

Now fractions with powers are cancelled, which gives .

Finally, the ratio of powers with the same exponents is replaced by powers of ratios, which leads to the equation , which is equivalent to . The transformations made allow us to introduce a new variable , which reduces the solution of the original exponential equation to the solution of the quadratic equation

I. V. Boikov, L. D. Romanova Collection of tasks for preparing for the exam. Part 1. Penza 2003.

An algebraic expression in the record of which, along with the operations of addition, subtraction and multiplication, also uses division into literal expressions, is called a fractional algebraic expression. Such are, for example, the expressions

We call an algebraic fraction an algebraic expression that has the form of a quotient of division of two integer algebraic expressions (for example, monomials or polynomials). Such are, for example, the expressions

the third of the expressions).

Identity transformations of fractional algebraic expressions are for the most part intended to represent them in the form algebraic fraction. To find a common denominator, the factorization of the denominators of fractions - terms is used in order to find their least common multiple. When reducing algebraic fractions, the strict identity of expressions can be violated: it is necessary to exclude the values ​​of quantities at which the factor by which the reduction is made vanishes.

Let us give examples of identical transformations of fractional algebraic expressions.

Example 1: Simplify an expression

All terms can be reduced to a common denominator (it is convenient to change the sign in the denominator of the last term and the sign in front of it):

Our expression is equal to one for all values ​​except these values, it is not defined and fraction reduction is illegal).

Example 2. Represent expression as an algebraic fraction

Solution. The expression can be taken as a common denominator. We find successively:

Exercises

1. Find the values ​​of algebraic expressions for the specified values ​​of the parameters:

2. Factorize.

Math-Calculator-Online v.1.0

The calculator performs the following operations: addition, subtraction, multiplication, division, working with decimals, extracting the root, raising to a power, calculating percentages, and other operations.

Solution:

How to use the math calculator

Key	Designation	Explanation
5	numbers 0-9	Arabic numerals. Enter natural integers, zero. To get a negative integer, press the +/- key
.	semicolon)	A decimal separator. If there is no digit before the dot (comma), the calculator will automatically substitute a zero before the dot. For example: .5 - 0.5 will be written
+	plus sign	Addition of numbers (whole, decimal fractions)
-	minus sign	Subtraction of numbers (whole, decimal fractions)
÷	division sign	Division of numbers (whole, decimal fractions)
X	multiplication sign	Multiplication of numbers (integers, decimals)
√	root	Extracting the root from a number. When you press the "root" button again, the root is calculated from the result. For example: square root of 16 = 4; square root of 4 = 2
x2	squaring	Squaring a number. When you press the "squaring" button again, the result is squared. For example: square 2 = 4; square 4 = 16
1/x	fraction	Output to decimals. In the numerator 1, in the denominator the input number
%	percent	Get a percentage of a number. To work, you must enter: the number from which the percentage will be calculated, the sign (plus, minus, divide, multiply), how many percent in numerical form, the "%" button
(	open bracket	An open parenthesis to set the evaluation priority. A closed parenthesis is required. Example: (2+3)*2=10
)	closed bracket	A closed parenthesis to set the evaluation priority. Availability required open bracket
±	plus minus	Changes sign to opposite
=	equals	Displays the result of the solution. Also, intermediate calculations and the result are displayed above the calculator in the "Solution" field.
←	deleting a character	Deletes the last character
FROM	reset	Reset button. Completely resets the calculator to "0"

The algorithm of the online calculator with examples

Addition.

Addition of whole natural numbers ( 5 + 7 = 12 )

Addition of whole natural and negative numbers ( 5 + (-2) = 3 )

Decimal addition fractional numbers { 0,3 + 5,2 = 5,5 }

Subtraction.

Subtraction of whole natural numbers ( 7 - 5 = 2 )

Subtraction of whole natural and negative numbers ( 5 - (-2) = 7 )

Subtraction of decimal fractional numbers ( 6.5 - 1.2 = 4.3 )

Multiplication.

Product of whole natural numbers ( 3 * 7 = 21 )

Product of whole natural and negative numbers ( 5 * (-3) = -15 )

Product of decimal fractional numbers ( 0.5 * 0.6 = 0.3 )

Division.

Division of whole natural numbers ( 27 / 3 = 9 )

Division of whole natural and negative numbers ( 15 / (-3) = -5 )

Division of decimal fractional numbers ( 6.2 / 2 = 3.1 )

Extracting the root from a number.

Extracting the root of an integer ( root(9) = 3 )

Extracting the root of decimals ( root(2.5) = 1.58 )

Extracting the root from the sum of numbers ( root(56 + 25) = 9 )

Extracting the root of the difference in numbers ( root (32 - 7) = 5 )

Squaring a number.

Squaring an integer ( (3) 2 = 9 )

Squaring decimals ( (2.2) 2 = 4.84 )

Convert to decimal fractions.

Calculating percentages of a number

Increase 230 by 15% ( 230 + 230 * 0.15 = 264.5 )

Decrease the number 510 by 35% ( 510 - 510 * 0.35 = 331.5 )

18% of the number 140 is ( 140 * 0.18 = 25.2 )

Convenient and simple online calculator fractions with detailed solution maybe:

• Add, subtract, multiply and divide fractions online,
• Receive turnkey solution fractions with a picture and it is convenient to transfer it.



The result of solving fractions will be here ...

0 1 2 3 4 5 6 7 8 9
Fraction sign "/" + - * :
_wipe Clear
Our online fraction calculator has fast input. To get the solution of fractions, for example, just write 1/2+2/7 into the calculator and press the " solve fractions". The calculator will write you detailed solution fractions and issue copy-friendly image.

The characters used for writing in the calculator

You can type an example for a solution both from the keyboard and using the buttons.

Features of the online fraction calculator

The fraction calculator can only perform operations with 2 simple fractions. They can be either correct (the numerator is less than the denominator) or incorrect (the numerator is greater than the denominator). The numbers in the numerator and denominators cannot be negative and greater than 999.
Our online calculator solves fractions and brings the answer to correct form- reduces the fraction and highlights the whole part, if necessary.

If you need to solve negative fractions, just use the minus properties. When multiplying and dividing negative fractions, minus by minus gives plus. That is, the product and division of negative fractions is equal to the product and division of the same positive ones. If one fraction is negative when multiplied or divided, then simply remove the minus, and then add it to the answer. When adding negative fractions, the result will be the same as if you added the same positive fractions. If you add one negative fraction, then this is the same as subtracting the same positive one.
When subtracting negative fractions, the result will be the same as if they were reversed and made positive. That is, a minus by a minus in this case gives a plus, and the sum does not change from a rearrangement of the terms. We use the same rules when subtracting fractions, one of which is negative.

To solve mixed fractions (fractions in which the whole part is highlighted), simply drive the whole part into a fraction. To do this, multiply the integer part by the denominator and add to the numerator.

If you need to solve 3 or more fractions online, then you should solve them one by one. First, count the first 2 fractions, then solve the next fraction with the answer received, and so on. Perform operations in turn for 2 fractions, and in the end you will get the correct answer.