Ministry of Education of the Republic of Belarus

educational institution

Gomel State University named after F. Skaryna"

Faculty of Mathematics

Department of MPM

Identical transformations of expressions and methods of teaching students how to perform them

Executor:

Student Starodubova A.Yu.

Scientific adviser:

Cand. physics and mathematics Sciences, Associate Professor Lebedeva M.T.

Gomel 2007

Introduction

1 The main types of transformations and stages of their study. Stages of mastering the application of transformations

Conclusion

Literature

Introduction

The simplest transformations of expressions and formulas, based on the properties of arithmetic operations, are performed in elementary school and in grades 5 and 6. The formation of skills and abilities to perform transformations takes place in the course of algebra. This is connected both with a sharp increase in the number and variety of transformations performed, and with the complication of activities to substantiate them and clarify the conditions of applicability, with the identification and study of generalized concepts of identity, identical transformation, equivalent transformation.

1. Main types of transformations and stages of their study. Stages of mastering the application of transformations

1. Beginnings of algebra

An unpartitioned system of transformations is used, represented by the rules for performing actions on one or both parts of the formula. The goal is to achieve fluency in performing tasks for solving the simplest equations, simplifying the formulas that define functions, in rationally performing calculations based on the properties of actions.

Typical examples:

Solve Equations:

a) ; b) ; in) .

Identity transformation (a); equivalent and identical (b).

2. Formation of skills for applying specific types of transformations

Conclusions: abbreviated multiplication formulas; transformations associated with exponentiation; transformations associated with various classes of elementary functions.

Organization of a holistic system of transformations (synthesis)

The goal is the formation of a flexible and powerful apparatus suitable for use in solving a variety of educational tasks.. The transition to this stage is carried out during the final repetition of the course in the course of comprehending the already known material learned in parts, for certain types of transformations, transformations of trigonometric expressions are added to the previously studied types. All these transformations can be called “algebraic” and “analytical” transformations include those based on the rules of differentiation and integration and transformations of expressions containing passages to the limit. The difference of this type is in the nature of the set that the variables run through in identities (certain sets of functions).

The identities under study are divided into two classes:

I are abbreviated multiplication identities valid in a commutative ring and identities

fair in the field.

II - identities connecting arithmetic operations and basic elementary functions.

2 Features of the organization of the task system in the study of identical transformations

The basic principle of organizing a system of tasks is to present them from simple to complex.

Exercise cycle- the combination in the sequence of exercises of several aspects of the study and methods of arranging the material. When studying identical transformations, the cycle of exercises is connected with the study of one identity, around which other identities are grouped, which are in a natural connection with it. The composition of the cycle, along with executive tasks, includes tasks, requiring recognition of the applicability of the considered identity. The identity under study is used to perform calculations on various numerical domains. Tasks in each cycle are divided into two groups. To first include tasks performed during the initial acquaintance with the identity. They serve as teaching material for several consecutive lessons, united by one topic.

Second group exercise connects the identity under study with various applications. This group does not form a compositional unity - the exercises here are scattered over various topics.

The described structures of the cycle refer to the stage of formation of skills for applying specific transformations.

At the stage of synthesis, cycles change, groups of tasks are combined towards complication and merging of cycles related to different identities, which increases the role of actions to recognize the applicability of one or another identity.

Example.

Identity task cycle:

I group of tasks:

a) present in the form of a product:

b) Check the correctness of the equality:

c) Expand the brackets in the expression:

.

d) Calculate:

e) Factorize:

e) simplify the expression:

.

The students have just got acquainted with the formulation of the identity, its recording in the form of an identity, and the proof.

Task a) is connected with fixing the structure of the identity under study, with establishing a connection with numerical sets (comparison of the sign structures of the identity and the expression being transformed; replacing a letter with a number in the identity). In the last example, it has yet to be reduced to the form under study. In the following examples (e and g), there is a complication caused by the applied role of identity and the complication of the sign structure.

Tasks of type b) are aimed at developing substitution skills on the . The role of task c) is similar.

Examples of type d), in which it is required to choose one of the transformation directions, completes the development of this idea.

Tasks of group I are focused on mastering the structure of the identity, the operation of substitution in the simplest, fundamentally most important cases, and the idea of ​​the reversibility of the transformations carried out by the identity. The enrichment of language means showing various aspects of identity is also very important. An idea about these aspects is given by the texts of tasks.

II group of tasks.

g) Using the identity for , factorize the polynomial .

h) Eliminate irrationality in the denominator of the fraction.

i) Prove that if is an odd number, then it is divisible by 4.

j) The function is given by the analytical expression

.

Get rid of the modulo sign by considering two cases: , .

l) Solve the equation .

These tasks are aimed at the fullest possible use and consideration of the specifics of this particular identity, suggest the formation of skills in using the identity under study for the difference of squares. The goal is to deepen the understanding of identity by considering its various applications in various situations, in combination with the use of material related to other topics of the mathematics course.

or .

Features of job cycles related to identities for elementary functions:

1) they are studied on the basis of functional material;

2) the identities of the first group appear later and are studied using the already formed skills for carrying out identical transformations.

The first group of tasks of the cycle should include tasks to establish a connection between these new numerical areas and the original area of ​​rational numbers.

Example.

Calculate:

;

.

The purpose of such tasks is to master the features of records, including symbols of new operations and functions, and to develop mathematical speech skills.

A significant part of the use of identity transformations associated with elementary functions falls on the solution of irrational and transcendental equations. Sequence of steps:

a) find a function φ for which the given equation f(x)=0 can be represented as:

b) make a substitution y=φ(x) and solve the equation

c) solve each of the equations φ(x)=y k , where y k is the set of roots of the equation F(y)=0.

When using the described method, step b) is often performed implicitly, without introducing a notation for φ(x). In addition, students often choose from among the various paths leading to finding an answer, to choose the one that leads to the algebraic equation faster and easier.

Example. Solve the equation 4 x -3*2=0.

2)(2 2) x -3*2 x =0 (step a)

(2 x) 2 -3*2 x =0; 2x(2x-3)=0; 2 x -3=0. (step b)

Example. Solve the equation:

a) 2 2x -3*2 x +2=0;

b) 2 2x -3*2 x -4=0;

c) 2 2x -3*2 x +1=0.

(Suggest for self-decision.)

Classification of tasks in cycles related to the solution of transcendental equations, including an exponential function:

1) equations that reduce to equations of the form a x \u003d y 0 and have a simple, general answer in form:

2) equations that reduce to equations of the form a x = a k , where k is an integer, or a x = b, where b≤0.

3) equations that reduce to equations of the form a x =y 0 and require an explicit analysis of the form in which the number y 0 is explicitly written.

Of great benefit are tasks in which identical transformations are used to plot graphs while simplifying formulas that define functions.

a) Plot the function y=;

b) Solve the equation lgx+lg(x-3)=1

c) on what set is the formula lg(x-5)+ lg(x+5)= lg(x 2 -25) an identity?

The use of identical transformations in calculations. (J. Mathematics at School, No. 4, 1983, p. 45)

Task number 1. The function is given by the formula y=0.3x 2 +4.64x-6. Find the function values ​​at x=1.2

y(1.2)=0.3*1.2 2 +4.64*1.2-6=1.2(0.3*1.2+4.64)-6=1.2(0 ,36+4.64)-6=1.2*5-6=0.

Task number 2. Calculate the length of the leg of a right triangle if the length of its hypotenuse is 3.6 cm, and the other leg is 2.16 cm.

Task number 3. What is the area of ​​a rectangular plot having dimensions a) 0.64m and 6.25m; b) 99.8m and 2.6m?

a) 0.64 * 6.25 \u003d 0.8 2 * 2.5 2 \u003d (0.8 * 2.5) 2;

b) 99.8*2.6=(100-0.2)2.6=100*2.6-0.2*2.6=260-0.52.

These examples make it possible to reveal the practical application of identical transformations. The student should be familiarized with the conditions for the feasibility of the transformation. (See diagrams).

-

image of a polynomial, where any polynomial fits into round contours. (Scheme 1)

-

the condition for the feasibility of converting the product of a monomial and an expression is given that allows conversion to the difference of squares. (scheme 2)

-

here, hatching means equal monomials and an expression is given that can be converted into a difference of squares. (Scheme 3)

-

an expression that allows the removal of a common factor.

To form students' skills in identifying conditions, you can use the following examples:

Which of the following expressions can be transformed by putting the common factor out of brackets:

2)

3) 0.7a 2 +0.2b 2;

5) 6,3*0,4+3,4*6,3;

6) 2x2 +3x2 +5y2;

7) 0,21+0,22+0,23.

Most of the calculations in practice do not satisfy the conditions of feasibility, so students need the skills to bring them to a form that allows for the calculation of transformations. In this case, the following tasks are appropriate:

when studying the removal of a common factor out of brackets:

this expression, if possible, transform into an expression, which is depicted by scheme 4:

4) 2a * a 2 * a 2;

5) 2n 4 +3n 6 +n 9 ;

8) 15ab 2 +5a 2 b;

10) 12,4*-1,24*0,7;

11) 4,9*3,5+1,7*10,5;

12) 10,8 2 -108;

13)

14) 5*2 2 +7*2 3 -11*2 4 ;

15) 2*3 4 -3*2 4 +6;

18) 3,2/0,7-1,8*

When forming the concept of “identical transformation”, it should be remembered that this means not only that the given and the resulting expression as a result of the transformation take equal values ​​for any values ​​of the letters included in it, but also that during the identical transformation we pass from the expression that determines one way of evaluating, to an expression that defines another way of evaluating the same value.

It is possible to illustrate scheme 5 (the rule for transforming the product of a monomial and a polynomial) with examples

0.5a(b+c) or 3.8(0.7+).

Exercises for learning to parenthesize the common factor:

Calculate the value of the expression:

a) 4.59*0.25+1.27*0.25+2.3-0.25;

b) a+bc at a=0.96; b=4.8; c=9.8.

c) a(a+c)-c(a+b) with a=1.4; b=2.8; c=5.2.

Let us illustrate with examples the formation of skills and abilities in calculations and identical transformations. (J. Mathematics at School, No. 5, 1984, p. 30)

1) skills and abilities are acquired faster and retained longer if their formation occurs on a conscious basis (the didactic principle of consciousness).

1) You can formulate the rule for adding fractions with the same denominators, or first, using specific examples, consider the essence of adding equal parts.

2) When factoring by taking the common factor out of brackets, it is important to see this common factor and then apply the distribution law. When performing the first exercises, it is useful to write each term of the polynomial as a product, one of the factors of which is common to all terms:

3a 3 -15a 2 b+5ab 2 = a3a 2 -a15ab+a5b 2 .

It is especially useful to do this when one of the monomials of the polynomial is taken out of the brackets:

II. First stage skill formation - mastering the skill (exercises are performed with detailed explanations and notes)

(the question of the sign is solved first)

Second phase- the stage of automating the skill by eliminating some intermediate operations

III. The strength of skills is achieved by solving examples that are diverse both in content and in form.

Topic: “Bracketing the common factor”.

1. Write down the missing multiplier instead of the polynomial:

2. Factorize so that before the brackets there is a monomial with a negative coefficient:

3. Factorize so that the polynomial in brackets has integer coefficients:

4. Solve the equation:

IV. The formation of skills is most effective in the case of oral performance of some intermediate calculations or transformations.

(orally);

V. The formed skills and abilities should be included in the previously formed system of knowledge, skills and abilities of students.

For example, when learning to factorize polynomials using abbreviated multiplication formulas, the following exercises are offered:

Multiply:

VI. The need for rational performance of calculations and transformations.

in) simplify the expression:

Rationality lies in the opening of brackets, because

VII. Converting expressions containing a degree.

№1011 (Alg.9) Simplify the expression:

№1012 (Alg.9) Take out the factor from under the root sign:

№1013 (Alg.9) Enter a factor under the root sign:

№1014 (Alg.9) Simplify the expression:

In all examples, preliminarily perform either factorization, or taking out a common factor, or “see” the corresponding reduction formula.

№1015 (Alg.9) Reduce the fraction:

Many students experience some difficulty in transforming expressions containing roots, in particular when investigating equality:

Therefore, either describe in detail expressions of the form or or go to a degree with a rational exponent.

№1018 (Alg.9) Find the value of the expression:

№1019 (Alg.9) Simplify the expression:

2.285 (Scanavi) Simplify expression

and then graph the function y for

No. 2.299 (Skanavi) Check the validity of equality:

The transformation of expressions containing a degree is a generalization of the acquired skills and abilities in the study of identical transformations of polynomials.

No. 2.320 (Skanavi) Simplify the expression:

In the Algebra 7 course, the following definitions are given.

Def. Two expressions whose corresponding values ​​are equal for the values ​​of the variables are said to be identically equal.

Def. Equality, true for any values ​​of the variables called. identity.

№94(Alg.7) Is the identity the equality:

a)

c)

d)

Description definition: The replacement of one expression by another, identically equal to it, is called an identical transformation or simply a transformation of an expression. Identical transformations of expressions with variables are performed based on the properties of operations on numbers.

№ (Alg.7) Among expressions

find those that are identically equal to .

Topic: "Identical transformations of expressions" (question methodology)

The first topic of "Algebra-7" - "Expressions and their transformations" helps to consolidate the computational skills acquired in grades 5-6, to systematize and generalize information about the transformations of expressions and solutions to equations.

Finding the values ​​of numerical and alphabetic expressions makes it possible to repeat with students the rules of action with rational numbers. The ability to perform arithmetic operations with rational numbers is the basis for the entire algebra course.

When considering transformations of expressions formally, operational skills remain at the same level that was achieved in grades 5-6.

However, here students rise to a new level in mastering the theory. The concepts of "identically equal expressions", "identity", "identical transformations of expressions" are introduced, the content of which will be constantly disclosed and deepened when studying the transformations of various algebraic expressions. It is emphasized that the basis of identical transformations is the properties of actions on numbers.

When studying the topic "Polynomials", formal-operational skills of identical transformations of algebraic expressions are formed. Abbreviated multiplication formulas contribute to the further process of forming skills to perform identical transformations of integer expressions, the ability to apply formulas both for abbreviated multiplication and for factoring polynomials is used not only in transforming integer expressions, but also in operations with fractions, roots, powers with a rational exponent .

In the 8th grade, the acquired skills of identical transformations are practiced on actions with algebraic fractions, square roots and expressions containing degrees with an integer exponent.

In the future, the methods of identical transformations are reflected in expressions containing a degree with a rational exponent.

A special group of identical transformations are trigonometric expressions and logarithmic expressions.

The mandatory learning outcomes for the algebra course in grades 7-9 include:

1) identical transformations of integer expressions

a) bracket opening and bracketing;

b) reduction of like terms;

c) addition, subtraction and multiplication of polynomials;

d) factorization of polynomials by taking the common factor out of brackets and abbreviated multiplication formulas;

e) factorization of a square trinomial.

"Mathematics at school" (B.U.M.) p.110

2) identical transformations of rational expressions: addition, subtraction, multiplication and division of fractions, as well as apply the listed skills when performing simple combined transformations [p. 111]

3) students should be able to perform transformations of simple expressions containing degrees and roots. (pp. 111-112)

The main types of tasks were considered, the ability to solve which allows the student to get a positive assessment.

One of the most important aspects of the methodology for studying identical transformations is the development by students of the goals of performing identical transformations.

1) - simplification of the numerical value of the expression

2) which of the transformations should be performed: (1) or (2) Analysis of these options is a motivation (preferably (1), because in (2) the definition area is narrowed)

3) Solve the equation:

Factorization in solving equations.

4) Calculate:

Let's apply the abbreviated multiplication formula:

(101-1) (101+1)=100102=102000

5) Find the value of the expression:

To find the value, multiply each fraction by the conjugate:

6) Plot the function graph:

Let's select the whole part: .

Error prevention when performing identical transformations can be obtained by varying examples of their execution. In this case, “small” techniques are worked out, which, as components, are included in a more voluminous transformation process.

For example:

Depending on the directions of the equation, several problems can be considered: from right to left multiplication of polynomials; from left to right - factorization. The left side is a multiple of one of the factors on the right side, and so on.

In addition to varying the examples, you can use the apology between identities and numerical equalities.

The next trick is to explain the identities.

To increase the interest of students, one can attribute the search for various ways to solve problems.

Lessons on the study of identical transformations will become more interesting if they are devoted to finding a solution to a problem .

For example: 1) reduce the fraction:

3) prove the "complex radical" formula

Consider:

Let's transform the right side of the equality:

-

sum of conjugate expressions. They could be multiplied and divided by the conjugate, but such an operation will lead us to a fraction whose denominator is the difference of the radicals.

Note that the first term in the first part of the identity is a number greater than the second, so you can square both parts:

Practical lesson number 3.

Topic: Identical transformations of expressions (question technique).

Literature: “Workshop on MPM”, pp. 87-93.

A sign of a high culture of calculations and identical transformations among students is a solid knowledge of the properties and algorithms of operations on exact and approximate values ​​and their skillful application; rational methods of calculations and transformations and their verification; the ability to substantiate the application of methods and rules of calculations and transformations, the automaticity of the skills of error-free execution of computational operations.

From what grade should students start working on developing these skills?

The line of identical transformations of expressions begins with the use of methods of rational calculation and begins with the use of methods of rational calculation of the values ​​of numerical expressions. (grade 5)

When studying such topics in a school mathematics course, special attention should be paid to them!

The conscious implementation of identical transformations by students is facilitated by the understanding of the fact that algebraic expressions do not exist on their own, but are inextricably linked with some numerical set, they are generalized records of numerical expressions. Analogies between algebraic and numerical expressions (and their transformations) are logically legitimate, their use in teaching helps to prevent students from making mistakes.

Identity transformations are not a separate topic of the school mathematics course, they are studied throughout the course of algebra and the beginning of mathematical analysis.

Mathematics program for grades 1-5 is a propaedeutic material for studying identical transformations of expressions with a variable.

In the course of algebra 7 cells. definitions of identity and identity transformations are introduced.

Def. Two expressions whose corresponding values ​​are equal for any values ​​of the variables, called. identically equal.

ODA. An equality that is true for any values ​​of the variables is called an identity.

The value of identity lies in the fact that it allows a given expression to be replaced by another identically equal to it.

Def. The replacement of one expression by another, identically equal to it, is called identity transformation or simply transformation expressions.

Identical transformations of expressions with variables are performed based on the properties of operations on numbers.

Equivalent transformations can be considered as the basis of identical transformations.

ODA. Two sentences, each of which is a logical consequence of the other, called. equivalent.

ODA. Sentence with variables A called. consequence of the sentence with variables B if the truth region B is a subset of the truth region A.

Another definition of equivalent sentences can be given: two sentences with variables are equivalent if their truth regions are the same.

a) B: x-1=0 over R; A: (x-1) 2 over R => A~B because truth regions (solutions) coincide (x=1)

b) A: x=2 over R; B: x 2 \u003d 4 over R => truth area A: x \u003d 2; truth region B: x=-2, x=2; because the truth region A is contained in B, then: x 2 =4 is a consequence of the sentence x=2.

The basis of identical transformations is the possibility of representing the same number in different forms. For example,

-

such a representation will help in studying the topic “basic properties of a fraction”.

Skills in performing identical transformations begin to form when solving examples similar to the following: “Find the numerical value of the expression 2a 3 + 3ab + b 2 with a \u003d 0.5, b \u003d 2/3”, which are offered to students in grade 5 and allow propaedeutics to be carried out concept of function.

When studying the formulas of abbreviated multiplication, attention should be paid to their deep understanding and strong assimilation. To do this, you can use the following graphic illustration:

(a+b) 2 =a 2 +2ab+b 2 (a-b) 2 =a 2 -2ab+b 2 a 2 -b 2 =(a-b)(a+b)

Question: How to explain to students the essence of the above formulas according to these drawings?

A common mistake is to confuse the expressions "squared sum" and "sum of squares". The teacher's indication that these expressions differ in the order of action does not seem significant, since students believe that these actions are performed on the same numbers and therefore the result does not change from changing the order of actions.

Task: Compose oral exercises to develop students' skills to accurately use the above formulas. How to explain how these two expressions are similar and how they differ from each other?

A wide variety of identical transformations makes it difficult for students to orientate themselves to the purpose for which they are being performed. Fuzzy knowledge of the purpose of performing transformations (in each specific case) negatively affects their awareness, and serves as a source of massive student errors. This suggests that explaining to students the goals of performing various identical transformations is an important part of the methodology for studying them.

Examples of motivations for identical transformations:

1. simplification of finding the numerical value of the expression;

2. choosing a transformation of the equation that does not lead to the loss of the root;

3. when performing a transformation, you can mark its area of ​​calculation;

4. the use of transformations in the calculation, for example, 99 2 -1=(99-1)(99+1);

To manage the decision process, it is important for the teacher to have the ability to give an accurate description of the essence of the mistake made by the student. Accurate characterization of the error is the key to the correct choice of subsequent actions taken by the teacher.

Examples of student errors:

1. performing multiplication: the student received -54abx 6 (7 cells);

2. performing exponentiation (3x 2) 3, the student received 3x 6 (7 cells);

3. transforming (m + n) 2 into a polynomial, the student received m 2 + n 2 (7 cells);

4. reducing the fraction the student received (8 cells);

5. performing subtraction: , the student writes down (8 cells)

6. Representing a fraction in the form of fractions, the student received: (8 cells);

7. extracting the arithmetic root, the student received x-1 (9 cells);

8. solving the equation (9 cells);

9. transforming the expression, the student receives: (9 cells).

Conclusion

The study of identical transformations is carried out in close connection with the numerical sets studied in one class or another.

At first, the student should be asked to explain each step of the transformation, to formulate the rules and laws that apply.

In identical transformations of algebraic expressions, two rules are used: substitution and replacement by equals. The most commonly used substitution, because formula counting is based on it, i.e. find the value of the expression a*b with a=5 and b=-3. Very often, students neglect parentheses when performing multiplication, believing that the multiplication sign is implied. For example, such record is possible: 5*-3.

Literature

1. A.I. Azarov, S.A. Barvenov "Functional and graphical methods for solving examination problems", Mn.. Aversev, 2004

2. O.N. Piryutko "Typical errors in centralized testing", Mn.. Aversev, 2006

3. A.I. Azarov, S.A. Barvenov "Tasks-traps on centralized testing", Mn.. Aversev, 2006

4. A.I. Azarov, S.A. Barvenov "Methods for solving trigonometric problems", Mn.. Aversev, 2005

Numeric and algebraic expressions. Expression conversion.

What is an expression in mathematics? Why are expression conversions necessary?

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

Let's say you have an evil example. Very large and very complex. Let's say you're good at math and you're not afraid of anything! Can you answer right away?

You'll have to decide this example. Sequentially, step by step, this example simplify. According to certain rules, of course. Those. do expression conversion. How successfully you carry out these transformations, so you are strong in mathematics. If you don't know how to do the right transformations, in mathematics you can't do nothing...

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

To begin with, let's find out what is an expression in math. What numeric expression and what is algebraic expression.

What is an expression in mathematics?

Expression in mathematics is a very broad concept. Almost everything we deal with 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 is also a mathematical expression. And a healthy fraction, and even one number - these are all mathematical expressions. The equation, for example, is:

5x + 2 = 12

consists of two mathematical expressions connected by an equals sign. One expression is on the left, the other is on the right.

In general terms, the term mathematical expression" is used, most often, in order not to mumble. They will ask you what an ordinary fraction is, for example? And how to answer ?!

Answer 1: "It's... m-m-m-m... such a thing ... in which ... Can I write a fraction better? Which one do you want?"

The second answer option: "An ordinary fraction is (cheerfully and joyfully!) mathematical expression , which consists of a numerator and a denominator!"

The second option is somehow more impressive, right?)

For this purpose, the phrase " mathematical expression "very good. Both correct and solid. But for practical application, you need to be well versed in specific kinds of expressions in mathematics .

The specific type is another matter. it quite another thing! Each type of mathematical expression has mine a set of rules and techniques that must be used in the decision. To work with fractions - one set. For working with trigonometric expressions - the second. For working with logarithms - the third. And so on. 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 master in the relevant sections.

Here we will master (or - repeat, as you like ...) two main types of mathematical expressions. Numeric expressions and algebraic expressions.

Numeric expressions.

What numeric 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 made up of numbers, brackets and signs of arithmetic operations is called a numeric expression.

7-3 is a numeric expression.

(8+3.2) 5.4 is also a numeric expression.

And this monster:

also a numeric expression, yes...

An ordinary number, a fraction, any calculation example without x's and other letters - all these are numerical expressions.

main feature numerical expressions in it no letters. None. Only numbers and mathematical icons (if necessary). It's simple, right?

And what can be done with numerical expressions? Numeric expressions can usually be counted. To do this, sometimes you have to open brackets, change signs, abbreviate, swap terms - i.e. do expression conversions. But more on that below.

Here we will deal with such a funny case when with a numerical expression you don't have to do anything. Well, nothing at all! This nice operation To do nothing)- is executed when the expression doesn't make sense.

When does a numeric expression not make sense?

Of course, if we see some kind of abracadabra in front of us, such as

then we won't do anything. Since it is not clear what to do with it. Some nonsense. Unless, to count the number of pluses ...

But there are outwardly quite decent expressions. For example this:

(2+3) : (16 - 2 8)

However, this expression is also doesn't make sense! For the simple reason that in the second brackets - if you count - you get zero. You can't divide by zero! This is a forbidden operation in mathematics. Therefore, there is no need to do anything with this expression either. For any task with such an expression, the answer will always be the same: "The expression doesn't make sense!"

To give such an answer, of course, I had to calculate what would be in brackets. And sometimes in brackets such a twist ... Well, there's nothing to be done about it.

There are not so many forbidden operations in mathematics. There is only one in this thread. Division by zero. Additional prohibitions arising in roots and logarithms are discussed in the relevant topics.

So, an idea of ​​what is numeric expression- got. concept numeric expression doesn't make sense- realized. Let's go further.

Algebraic expressions.

If letters appear in a 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; ...

Such expressions are also called literal expressions. Or expressions with variables. It's practically the same thing. Expression 5a +c, for example - both literal and algebraic, and expression with variables.

concept algebraic expression - wider than numerical. It includes and all numeric expressions. Those. a numeric expression is also an algebraic expression, only without the letters. Every herring is a fish, but not every fish is a herring...)

Why literal- clear. Well, since there are letters ... Phrase expression with variables also not very perplexing. If you understand that numbers are hidden under the letters. All sorts of numbers can be hidden under the letters ... And 5, and -18, and whatever you like. That is, a letter can replace for different numbers. That's why the letters are called variables.

In the expression y+5, for example, at- variable. Or just say " variable", without the word "value". Unlike the five, which is a constant value. Or simply - constant.

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

In arithmetic, one can write that

But if we write a similar equality through algebraic expressions:

a + b = b + a

we will decide immediately all questions. For all numbers stroke. For an infinite number of things. Because under the letters a and b implied all numbers. And not only numbers, but even other mathematical expressions. This is how algebra works.

When does an algebraic expression make no sense?

Everything is clear about the numerical expression. You can't divide by zero. And with letters, is it possible to find out what we are dividing by ?!

Let's take the following variable expression as an example:

2: (a - 5)

Does it make sense? But who knows him? a- any number...

Any, any... But there is one meaning a, for which this expression exactly doesn't make sense! And what is that number? Yes! It's 5! If the variable a replace (they say - "substitute") with the number 5, in parentheses, zero will turn out. which cannot be divided. So it turns out that our expression doesn't make sense, if a = 5. But for other values a does it make sense? Can you substitute other numbers?

Of course. In such cases, it is simply said that the expression

2: (a - 5)

makes sense for any value a, except a = 5 .

The entire set of numbers can substitute into the given expression is called valid range this expression.

As you can see, there is nothing tricky. We look at the expression with variables, and think: at what value of the variable is the forbidden operation obtained (division by zero)?

And then be sure to look at the question of the assignment. What are they asking?

doesn't make sense, our forbidden value will be the answer.

If they ask at what value of the variable the expression has the meaning(feel the difference!), the answer will be all other numbers except for the forbidden.

Why do we need the meaning of the expression? He is there, he is not... 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 range of valid values ​​or the scope of a function. Without this, you will not be able to solve serious equations or inequalities at all. Like this.

Expression conversion. Identity transformations.

We got acquainted with numerical and algebraic expressions. Understand what the phrase "the expression does not make sense" means. Now we need to figure out what expression conversion. The answer is simple, outrageously.) This is any action with an expression. And that's it. You have been doing these transformations since the first class.

Take the cool numerical expression 3+5. How can it be converted? Yes, very easy! Calculate:

This calculation will be the transformation of the expression. You can write the same expression in a different way:

We didn't count anything here. Just write down the expression in a different form. This will also be a transformation of the expression. It can be written like this:

And this, too, is the transformation of an expression. You can make as many of these transformations as you like.

Any action on an expression any writing it in a different form is called an expression transformation. And all things. Everything is very simple. But there is one thing here very important rule. So important that it can safely be called main rule all mathematics. Breaking this rule inevitably leads to errors. Do we understand?)

Let's say we've transformed our expression arbitrarily, like this:

Transformation? Of course. We wrote the expression in a different form, what is wrong here?

It's not like that.) The fact is that the transformations "whatever" mathematics is not interested at all.) All mathematics is built on transformations in which the appearance changes, but the essence of the expression does not change. Three plus five can be written in any form, but it must be eight.

transformations, expressions that do not change the essence called identical.

Exactly identical transformations and allow us, step by step, to turn a complex example into a simple expression, keeping essence of the example. If we make a mistake in the chain of transformations, we will make a NOT identical transformation, then we will decide another example. With other answers that are not related to the correct ones.)

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

I gave an example with a numerical expression 3 + 5 for clarity. In algebraic expressions, identical transformations are given by formulas and rules. Let's say there is a formula in algebra:

a(b+c) = ab + ac

So, in any example, we can instead of the expression a(b+c) feel free to write an expression ab+ac. And vice versa. it identical transformation. Mathematics gives us a choice of these two expressions. And which one to write depends on the specific example.

Another example. One of the most important and necessary transformations is the basic property of a fraction. You can see more details at the link, but here I just remind the rule: if the numerator and denominator of a fraction are multiplied (divided) by the same number, or an expression that is not equal to zero, the fraction will not change. Here is an example of identical transformations for this property:

As you probably guessed, this chain can be continued indefinitely...) A very important property. It is it that allows you to turn all sorts of example monsters into white and fluffy.)

There are many formulas defining identical transformations. But the most important - quite a reasonable amount. One of the basic transformations is factorization. It is used in all mathematics - from elementary to advanced. Let's start with him. in the next lesson.)

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Often we hear this unpleasant phrase: "simplify the expression." Usually, in this case, we have some kind of monster like this:

“Yes, much easier,” we say, but such an answer usually does not work.

Now I will teach you not to be afraid of any such tasks.

Moreover, at the end of the lesson, you yourself will simplify this example to a (just!) ordinary number (yes, to hell with these letters).

But before you start this lesson, you need to be able to deal with fractions and factorize polynomials.

Therefore, if you have not done this before, be sure to master the topics "" and "".

Read? If yes, then you are ready.

Let's go! (Let's go!)

Basic Expression Simplification Operations

Now we will analyze the main techniques that are used to simplify expressions.

The simplest of them is

1. Bringing similar

What are similar? You went through this in 7th grade, when letters first appeared in math instead of numbers.

Similar are terms (monomials) with the same letter part.

For example, in the sum, like terms are and.

Remembered?

Bring similar- means to add several similar terms with each other and get one term.

But how can we put letters together? - you ask.

This is very easy to understand if you imagine that the letters are some kind of objects.

For example, the letter is a chair. Then what is the expression?

Two chairs plus three chairs, how much will it be? That's right, chairs: .

Now try this expression:

In order not to get confused, let different letters denote different objects.

For example, - this is (as usual) a chair, and - this is a table.

chairs tables chair tables chairs chairs tables

The numbers by which the letters in such terms are multiplied are called coefficients.

For example, in the monomial the coefficient is equal. And he is equal.

So, the rule for bringing similar:

Examples:

Bring similar:

Answers:

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

2. Factorization

This is usually the most important part in simplifying expressions.

After you have given similar ones, most often the resulting expression is needed factorize, i.e. represent as a product.

Especially this important in fractions: because in order to reduce the fraction, the numerator and denominator must be expressed as a product.

You went through the detailed methods of factoring expressions in the topic "", so here you just have to remember what you have learned.

To do this, solve a few examples (you need to factorize)

Examples:

Solutions:

3. Fraction reduction.

Well, what could be nicer than to cross out part of the numerator and denominator, and throw them out of your life?

That's the beauty of abbreviation.

It's simple:

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

This rule follows from the basic property of a fraction:

That is, the essence of the reduction operation is that We divide the numerator and denominator of a fraction by the same number (or by the same expression).

To reduce a fraction, you need:

1) numerator and denominator factorize

2) if the numerator and denominator contain common factors, they can be deleted.

Examples:

The principle, I think, is clear?

I would like to draw your attention to one typical mistake in abbreviation. Although this topic is simple, but many people do everything wrong, not realizing that cut- this means divide numerator and denominator by the same number.

No abbreviations if the numerator or denominator is the sum.

For example: you need to simplify.

Some do this: which is absolutely wrong.

Another example: reduce.

The "smartest" will do this:

Tell me what's wrong here? It would seem: - this is a multiplier, so you can reduce.

But no: - this is a factor of only one term in the numerator, but the numerator itself as a whole is not decomposed into factors.

Here is another example: .

This expression is decomposed into factors, which means that you can reduce, that is, divide the numerator and denominator by, and then by:

You can immediately divide by:

To avoid such mistakes, remember an easy way to determine if an expression is factored:

The arithmetic operation that is performed last when calculating the value of the expression is the "main".

That is, if you substitute some (any) numbers instead of letters, and try to calculate the value of the expression, then if the last action is multiplication, then we have a product (the expression is decomposed into factors).

If the last action is addition or subtraction, this means that the expression is not factored (and therefore cannot be reduced).

To fix it yourself, a few examples:

Examples:

Solutions:

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

Adding and subtracting ordinary fractions is a well-known operation: we look for a common denominator, multiply each fraction by the missing factor and add / subtract the numerators.

Let's remember:

Answers:

1. The denominators and are coprime, that is, they do not have common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:

2. Here the common denominator is:

3. Here, first of all, we turn mixed fractions into improper ones, and then - according to the usual scheme:

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

Let's start simple:

a) Denominators do not contain letters

Here everything is the same as with ordinary numerical fractions: we find a common denominator, multiply each fraction by the missing factor and add / subtract the numerators:

now in the numerator you can bring similar ones, if any, and factor them:

Try it yourself:

Answers:

b) Denominators contain letters

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

First of all, we determine the common factors;

Then we write out all the common factors once;

and multiply them by all other factors, not common ones.

To determine the common factors of the denominators, we first decompose them into simple factors:

We emphasize the common factors:

Now we write out the common factors once and add to them all non-common (not underlined) factors:

This is the common denominator.

Let's get back to the letters. The denominators are given in exactly the same way:

We decompose the denominators into factors;

determine common (identical) multipliers;

write out all the common factors once;

We multiply them by all other factors, not common ones.

So, in order:

1) decompose the denominators into factors:

2) determine the common (identical) factors:

3) write out all the common factors once and multiply them by all the other (not underlined) factors:

So the common denominator is here. The first fraction must be multiplied by, the second - by:

By the way, there is one trick:

For example: .

We see the same factors in the denominators, only all with different indicators. The common denominator will be:

to the extent

to the extent

to the extent

in degree.

Let's complicate the task:

How to make fractions have the same denominator?

Let's remember the basic property of a fraction:

Nowhere is it said that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because it's not true!

See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example, . What has been learned?

So, another unshakable rule:

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

But what do you need to multiply to get?

Here on and multiply. And multiply by:

Expressions that cannot be factorized will be called "elementary factors".

For example, is an elementary factor. - too. But - no: it is decomposed into factors.

What about expression? Is it elementary?

No, because it can be factorized:

(you already read about factorization in the topic "").

So, the elementary factors into which you decompose an expression with letters are an analogue of the simple factors into which you decompose numbers. And we will do the same with them.

We see that both denominators have a factor. It will go to the common denominator in the power (remember why?).

The multiplier is elementary, and they do not have it in common, which means that the first fraction will simply have to be multiplied by it:

Another example:

Solution:

Before multiplying these denominators in a panic, you need to think about how to factor them? Both of them represent:

Excellent! Then:

Another example:

Solution:

As usual, we factorize the denominators. In the first denominator, we simply put it out of brackets; in the second - the difference of squares:

It would seem that there are no common factors. But if you look closely, they are already so similar ... And the truth is:

So let's write:

That is, it turned out like this: inside the bracket, we swapped the terms, and at the same time, the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.

Now we bring to a common denominator:

Got it? Now let's check.

Tasks for independent solution:

Answers:

5. Multiplication and division of fractions.

Well, the hardest part is now over. And ahead of us is the simplest, but at the same time the most important:

Procedure

What is the procedure for calculating a numeric expression? Remember, considering the value of such an expression:

Did you count?

It should work.

So, I remind you.

The first step is to calculate the degree.

The second is multiplication and division. If there are several multiplications and divisions at the same time, you can do them in any order.

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

But: the parenthesized expression is evaluated out of order!

If several brackets are multiplied or divided by each other, we first evaluate the expression in each of the brackets, and then multiply or divide them.

What if there are other parentheses inside the brackets? Well, let's think: some expression is written inside the brackets. What is the first thing to do when evaluating an expression? That's right, calculate brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.

So, the order of actions for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):

Okay, it's all simple.

But that's not the same as an expression with letters, is it?

No, it's the same! Only instead of arithmetic operations it is necessary to do algebraic operations, that is, the operations described in the previous section: bringing similar, adding fractions, reducing fractions, and so on. The only difference will be the action of factoring polynomials (we often use it when working with fractions). Most often, for factorization, you need to use i or simply take the common factor out of brackets.

Usually our goal is to represent an expression as a product or quotient.

For example:

Let's simplify the expression.

1) First we simplify the expression in brackets. There we have the difference of fractions, and our goal is to represent it as a product or quotient. So, we bring the fractions to a common denominator and add:

It is impossible to simplify this expression further, all factors here are elementary (do you still remember what this means?).

2) We get:

Multiplication of fractions: what could be easier.

3) Now you can shorten:

OK it's all over Now. Nothing complicated, right?

Another example:

Simplify the expression.

First, try to solve it yourself, and only then look at the solution.

Solution:

First of all, let's define the procedure.

First, let's add the fractions in brackets, instead of two fractions, one will turn out.

Then we will do the division of fractions. Well, we add the result with the last fraction.

I will schematically number the steps:

Finally, I will give you two useful tips:

1. If there are similar ones, they must be brought immediately. At whatever moment we have similar ones, it is advisable to bring them right away.

2. The same goes for reducing fractions: as soon as an opportunity arises to reduce, it must be used. The exception is fractions that you add or subtract: if they now have the same denominators, then the reduction should be left for later.

Here are some tasks for you to solve on your own:

And promised at the very beginning:

Answers:

Solutions (brief):

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

Now on to learning!

EXPRESSION CONVERSION. SUMMARY AND BASIC FORMULA

Basic simplification operations:

• Bringing similar: to add (reduce) like terms, you need to add their coefficients and assign the letter part.
• Factorization: taking the common factor out of brackets, applying, etc.
• Fraction reduction: the numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, from which the value of the fraction does not change.
  1) numerator and denominator factorize
  2) if there are common factors in the numerator and denominator, they can be crossed out.

  IMPORTANT: only multipliers can be reduced!

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

Well, the topic is over. If you are reading these lines, then you are very cool.

Because only 5% of people are able to master something on their own. And if you have read to the end, then you are in the 5%!

Now the most important thing.

You've figured out the theory on this topic. And, I repeat, it's ... it's just super! You are already better than the vast majority of your peers.

The problem is that this may not be enough ...

For what?

For the successful passing of the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.

I will not convince you of anything, I will just say one thing ...

People who have received a good education earn much more than those who have not received it. This is statistics.

But this is not the main thing.

The main thing is that they are MORE HAPPY (there are such studies). Perhaps because much more opportunities open up before them and life becomes brighter? Don't know...

But think for yourself...

What does it take to be sure to be better than others on the exam and be ultimately ... happier?

FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.

On the exam, you will not be asked theory.

You will need solve problems on time.

And, if you haven’t solved them (LOTS!), you will definitely make a stupid mistake somewhere or simply won’t make it in time.

It's like in sports - you need to repeat many times to win for sure.

Find a collection anywhere you want necessarily with solutions, detailed analysis and decide, decide, decide!

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The numbers and expressions that make up the original expression can be replaced by expressions that are identically equal to them. Such a transformation of the original expression leads to an expression that is identically equal to it.

For example, in the expression 3+x, the number 3 can be replaced by the sum 1+2 , which results in the expression (1+2)+x , which is identically equal to the original expression. Another example: in the expression 1+a 5 the degree of a 5 can be replaced by a product identically equal to it, for example, of the form a·a 4 . This will give us the expression 1+a·a 4 .

This transformation is undoubtedly artificial, and is usually a preparation for some further transformation. For example, in the sum 4·x 3 +2·x 2 , taking into account the properties of the degree, the term 4·x 3 can be represented as a product 2·x 2 ·2·x . After such a transformation, the original expression will take the form 2·x 2 ·2·x+2·x 2 . Obviously, the terms in the resulting sum have a common factor 2 x 2, so we can perform the following transformation - parentheses. After it, we will come to the expression: 2 x 2 (2 x+1) .

Adding and subtracting the same number

Another artificial transformation of an expression is the addition and subtraction of the same number or expression at the same time. Such a transformation is identical, since it is, in fact, equivalent to adding zero, and adding zero does not change the value.

Consider an example. Let's take the expression x 2 +2 x . If you add one to it and subtract one, then this will allow you to perform another identical transformation in the future - select the square of the binomial: x 2 +2 x=x 2 +2 x+1−1=(x+1) 2 −1.

Bibliography.

• Algebra: textbook for 7 cells. 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: textbook for 8 cells. 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.
• Mordkovich A. G. Algebra. 7th grade. At 2 pm Part 1. A textbook for students of educational institutions / A. G. Mordkovich. - 17th ed., add. - M.: Mnemozina, 2013. - 175 p.: ill. ISBN 978-5-346-02432-3.

Basic properties of addition and multiplication of numbers.

Commutative property of addition: when the terms are rearranged, the value of the sum does not change. For any numbers a and b, the equality is true

The 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 to the first number. For any numbers a, b and c the equality is true

Commutative property of multiplication: permutation of factors does not change the value of the product. For any numbers a, b and c, the equality is true

The associative property of multiplication: in order to multiply the product of two numbers by a third number, you can multiply the first number by the product of the second and third.

For any numbers a, b and c, the equality is true

Distributive property: To multiply a number by a sum, you can multiply that number by each term and add the results. For any numbers a, b and c the equality is true

It follows from the commutative and associative properties of addition that in any sum you can rearrange the terms as you like and combine them into groups in an arbitrary way.

Example 1 Let's calculate the sum 1.23+13.5+4.27.

To do this, it is convenient to combine the first term with the third. We get:

1,23+13,5+4,27=(1,23+4,27)+13,5=5,5+13,5=19.

It follows from the commutative and associative properties of multiplication: in any product, you can rearrange the factors in any way and arbitrarily combine them into groups.

Example 2 Let's find the value of the product 1.8 0.25 64 0.5.

Combining the first factor with the fourth, and the second with the third, we will have:

1.8 0.25 64 0.5 \u003d (1.8 0.5) (0.25 64) \u003d 0.9 16 \u003d 14.4.

The distribution property is also valid when the number is multiplied by the sum of three or more terms.

For example, for any numbers a, b, c and d, the equality is true

a(b+c+d)=ab+ac+ad.

We know that subtraction can be replaced by addition by adding to the minuend the opposite number to the subtrahend:

This allows a numerical expression of the form a-b to be considered the sum of numbers a and -b, a numerical expression of the form a + b-c-d to be considered the sum of numbers a, b, -c, -d, etc. The considered properties of actions are also valid for such sums.

Example 3 Let's find the value of the expression 3.27-6.5-2.5+1.73.

This expression is the sum of the numbers 3.27, -6.5, -2.5 and 1.73. Applying the addition properties, we get: 3.27-6.5-2.5+1.73=(3.27+1.73)+(-6.5-2.5)=5+(-9) = -four.

Example 4 Let's calculate the product 36·().

The multiplier can be thought of as the sum of the numbers and -. Using the distributive property of multiplication, we get:

36()=36-36=9-10=-1.

Identities

Definition. Two expressions whose corresponding values ​​are equal for any values ​​of the variables are said to be identically equal.

Definition. An equality that is true for any values ​​of the variables is called an identity.

Let's find the values ​​of the expressions 3(x+y) and 3x+3y for x=5, y=4:

3(x+y)=3(5+4)=3 9=27,

3x+3y=3 5+3 4=15+12=27.

We got the same result. It follows from the distributive property that, in general, for any values ​​of the variables, the corresponding values ​​of the expressions 3(x+y) and 3x+3y are equal.

Consider now the expressions 2x+y and 2xy. For x=1, y=2 they take equal values:

However, you can specify x and y values ​​such that the values ​​of these expressions are not equal. For example, if x=3, y=4, then

The expressions 3(x+y) and 3x+3y are identically equal, but the expressions 2x+y and 2xy are not identically equal.

The equality 3(x+y)=x+3y, true for any values ​​of x and y, is an identity.

True numerical equalities are also considered identities.

So, identities are equalities expressing the main properties of actions on numbers:

a+b=b+a, (a+b)+c=a+(b+c),

ab=ba, (ab)c=a(bc), a(b+c)=ab+ac.

Other examples of identities can be given:

a+0=a, a+(-a)=0, a-b=a+(-b),

a 1=a, a (-b)=-ab, (-a)(-b)=ab.

Identity transformations of expressions

The replacement of one expression by another, identically equal to it, is called an identical transformation or simply a transformation of an expression.

Identical transformations of expressions with variables are performed based on the properties of operations on numbers.

To find the value of the expression xy-xz given the values ​​x, y, z, you need to perform three steps. For example, with x=2.3, y=0.8, z=0.2 we get:

xy-xz=2.3 0.8-2.3 0.2=1.84-0.46=1.38.

This result can be obtained in only two steps, using the expression x(y-z), which is identically equal to the expression xy-xz:

xy-xz=2.3(0.8-0.2)=2.3 0.6=1.38.

We have simplified the calculations by replacing the expression xy-xz with the identically equal expression x(y-z).

Identity transformations of expressions are widely used in calculating the values ​​of expressions and solving other problems. Some identical transformations have already been performed, for example, the reduction of similar terms, the opening of brackets. Recall the rules for performing these transformations:

to bring like terms, you need to add their coefficients and multiply the result by the common letter part;

if there is a plus sign in front of the brackets, then the brackets can be omitted, retaining the sign of each term enclosed in brackets;

if there is a minus sign before the brackets, then the brackets can be omitted by changing the sign of each term enclosed in brackets.

Example 1 Let's add like terms in the sum 5x+2x-3x.

We use the rule for reducing like terms:

5x+2x-3x=(5+2-3)x=4x.

This transformation is based on the distributive property of multiplication.

Example 2 Let's expand the brackets in the expression 2a+(b-3c).

Applying the rule for opening brackets preceded by a plus sign:

2a+(b-3c)=2a+b-3c.

The performed transformation is based on the associative property of addition.

Example 3 Let's expand the brackets in the expression a-(4b-c).

Let's use the rule for expanding brackets preceded by a minus sign:

a-(4b-c)=a-4b+c.

The performed transformation is based on the distributive property of multiplication and the associative property of addition. Let's show it. Let's represent the second term -(4b-c) in this expression as a product (-1)(4b-c):

a-(4b-c)=a+(-1)(4b-c).

Applying these properties of actions, we get:

a-(4b-c)=a+(-1)(4b-c)=a+(-4b+c)=a-4b+c.