Expression conversion. Detailed Theory (2020)

(1) a m ⋅ a n = a m + n

Example:

$$(a^2) \cdot (a^5) = (a^7)$$ (2) a m a n = a m − n

Example:

$$\frac(((a^4)))(((a^3))) = (a^(4 – 3)) = (a^1) = a$$ (3) (a ⋅ b) n = a n ⋅ b n

Example:

$$((a \cdot b)^3) = (a^3) \cdot (b^3)$$ (4) (a b) n = a n b n

Example:

$$(\left((\frac(a)(b)) \right)^8) = \frac(((a^8)))(((b^8)))$$ (5) (a m ) n = a m ⋅ n

Example:

$$(((a^2))^5) = (a^(2 \cdot 5)) = (a^(10))$$ (6) a − n = 1 a n

Examples:

$$(a^( – 2)) = \frac(1)(((a^2)));\;\;\;\;(a^( – 1)) = \frac(1)(( (a^1))) = \frac(1)(a).$$

Square root properties:

(1) a b = a ⋅ b , for a ≥ 0 , b ≥ 0

Example:

18 = 9 ⋅ 2 = 9 ⋅ 2 = 3 2

(2) a b = a b , for a ≥ 0 , b > 0

Example:

4 81 = 4 81 = 2 9

(3) (a) 2 = a , for a ≥ 0

Example:

(4) a 2 = | a | for any a

Examples:

(− 3) 2 = | − 3 | = 3 , 4 2 = | 4 | = 4 .

Rational and irrational numbers

Rational numbers are numbers that can be represented as a common fraction m n where m is an integer (ℤ = 0, ± 1, ± 2, ± 3 …), n is a natural number (ℕ = 1, 2, 3, 4 …).

Examples of rational numbers:

1 2 ; − 9 4 ; 0,3333 … = 1 3 ; 8 ; − 1236.

Irrational numbers - numbers that cannot be represented as an ordinary fraction m n, these are infinite non-periodic decimal fractions.

Examples of irrational numbers:

e = 2.71828182845…

π = 3.1415926…

2 = 1,414213562…

3 = 1,7320508075…

Simply put, irrational numbers are numbers that contain the square root sign in their notation. But not everything is so simple. Some rational numbers disguise themselves as irrational ones, for example, the number 4 contains a square root sign in its notation, but we are well aware that we can simplify the notation 4 = 2. This means that the number 4 is a rational number.

Similarly, the number 4 81 = 4 81 = 2 9 is a rational number.

Some problems require you to determine which numbers are rational and which are irrational. The task is to understand which numbers are irrational and which are disguised as them. To do this, you need to be able to perform the operations of taking the factor out from under the square root sign and introducing the factor under the root sign.

Insertion and removal of the factor for the sign of the square root

By taking the factor out of the square root sign, you can significantly simplify some mathematical expressions.

Example:

Simplify expression 2 8 2 .

1 way (taking out the multiplier from under the root sign): 2 8 2 = 2 4 ⋅ 2 2 = 2 4 ⋅ 2 2 = 2 ⋅ 2 = 4

Method 2 (introducing a multiplier under the root sign): 2 8 2 = 2 2 8 2 = 4 ⋅ 8 2 = 4 ⋅ 8 2 = 16 = 4

Abbreviated multiplication formulas (FSU)

sum square

(1) (a + b) 2 = a 2 + 2 a b + b 2

Example:

(3 x + 4 y) 2 = (3 x) 2 + 2 ⋅ 3 x ⋅ 4 y + (4 y) 2 = 9 x 2 + 24 x y + 16 y 2

The square of the difference

(2) (a − b) 2 = a 2 − 2 a b + b 2

Example:

(5 x − 2 y) 2 = (5 x) 2 − 2 ⋅ 5 x ⋅ 2 y + (2 y) 2 = 25 x 2 − 20 x y + 4 y 2

Sum of squares does not factor

Difference of squares

(3) a 2 − b 2 = (a − b) (a + b)

Example:

25 x 2 - 4 y 2 = (5 x) 2 - (2 y) 2 = (5 x - 2 y) (5 x + 2 y)

sum cube

(4) (a + b) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3

Example:

(x + 3 y) 3 = (x) 3 + 3 ⋅ (x) 2 ⋅ (3 y) + 3 ⋅ (x) ⋅ (3 y) 2 + (3 y) 3 = x 3 + 3 ⋅ x 2 ⋅ 3 y + 3 ⋅ x ⋅ 9 y 2 + 27 y 3 = x 3 + 9 x 2 y + 27 x y 2 + 27 y 3

difference cube

(5) (a − b) 3 = a 3 − 3 a 2 b + 3 a b 2 − b 3

Example:

(x 2 − 2 y) 3 = (x 2) 3 − 3 ⋅ (x 2) 2 ⋅ (2 y) + 3 ⋅ (x 2) ⋅ (2 y) 2 − (2 y) 3 = x 2 ⋅ 3 − 3 ⋅ x 2 ⋅ 2 ⋅ 2 y + 3 ⋅ x 2 ⋅ 4 y 2 − 8 y 3 = x 6 − 6 x 4 y + 12 x 2 y 2 − 8 y 3

Sum of cubes

(6) a 3 + b 3 = (a + b) (a 2 − a b + b 2)

Example:

8 + x 3 = 2 3 + x 3 = (2 + x) (2 2 − 2 ⋅ x + x 2) = (x + 2) (4 − 2 x + x 2)

Difference of cubes

(7) a 3 − b 3 = (a − b) (a 2 + a b + b 2)

Example:

x 6 - 27 y 3 = (x 2) 3 - (3 y) 3 = (x 2 - 3 y) ((x 2) 2 + (x 2) (3 y) + (3 y) 2) = ( x 2 − 3 y) (x 4 + 3 x 2 y + 9 y 2)

Standard form of number

In order to understand how to bring an arbitrary rational number to the standard form, you need to know what the first significant digit of the number is.

The first significant digit of a number call it the first non-zero digit on the left.

Examples:
2 5 ; 3, 05; 0 , 143 ; 0 , 00 1 2 . The first significant digit is highlighted in red.

To convert a number to standard form:

Shift the comma so that it is immediately after the first significant digit.
Multiply the resulting number by 10 n, where n is a number, which is defined as follows:
n > 0 if the comma was shifted to the left (multiplying by 10 n indicates that the comma should actually be to the right);
n< 0 , если запятая сдвигалась вправо (умножение на 10 n , указывает, что на самом деле запятая должна стоять левее);
the absolute value of the number n is equal to the number of digits by which the comma was shifted.

Examples:

25 = 2 , 5 ← , = 2,5 ⋅ 10 1

The comma has moved to the left by 1 digit. Since the decimal point is shifted to the left, the exponent is positive.

Already brought to the standard form, you do not need to do anything with it. It can be written as 3.05 ⋅ 10 0 , but since 10 0 = 1, we leave the number in its original form.

0,143 = 0, 1 → , 43 = 1,43 ⋅ 10 − 1

The comma has moved to the right by 1 digit. Since the decimal point is shifted to the right, the exponent is negative.

− 0,0012 = − 0, 0 → 0 → 1 → , 2 = − 1,2 ⋅ 10 − 3

The comma has moved three places to the right. Since the decimal point is shifted to the right, the exponent is negative.

Algebraic expression

an expression made up of letters and numbers connected by the signs of the operations of addition, subtraction, multiplication, division, raising to an integer power and extracting the root (the exponents and the root must be constant numbers). A. in. is called rational with respect to some of the letters included in it if it does not contain them under the root extraction sign, for example

rational with respect to a, b and c. A. in. is called an integer with respect to some letters if it does not contain division by expressions containing these letters, for example 3a / c + bc 2 - 3ac / 4 is integer with respect to a and b. If some of the letters (or all) are considered variables, then A. c. is an algebraic function.

Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

See what "Algebraic Expression" is in other dictionaries:

An expression made up of letters and numbers connected by signs of algebraic operations: addition, subtraction, multiplication, division, raising to a power, extracting a root ... Big Encyclopedic Dictionary

algebraic expression- - Topics oil and gas industry EN algebraic expression ... Technical Translator's Handbook

An algebraic expression is one or more algebraic quantities (numbers and letters) interconnected by signs of algebraic operations: addition, subtraction, multiplication and division, as well as extracting the root and raising to an integer ... ... Wikipedia

algebraic expression- algebrinė išraiška statusas T sritis fizika atitikmenys: engl. algebraic expression vok. algebraischer Ausdruck, m rus. algebraic expression, n pranc. expression algebrique, f … Fizikos terminų žodynas

An expression made up of letters and numbers connected by the signs of algebras. actions: addition, subtraction, multiplication, division, exponentiation, root extraction ... Natural science. encyclopedic Dictionary

An algebraic expression with respect to a given variable, in contrast to a transcendental one, is an expression that does not contain other functions of a given quantity, except for sums, products or powers of this quantity, moreover, terms ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron

EXPRESSION, expressions, cf. 1. Action according to Ch. express express. I can't find words to express my gratitude. 2. more often than not The embodiment of an idea in the forms of some kind of art (philosophical). Only a great artist is able to create such an expression, ... ... Explanatory Dictionary of Ushakov

An equation obtained by equating two algebraic expressions (See Algebraic Expression). A. y. with one unknown is called fractional if the unknown is included in the denominator, and irrational if the unknown is included under ... ... Great Soviet Encyclopedia

EXPRESSION- a primary mathematical concept, which means a record of letters and numbers connected by signs of arithmetic operations, while brackets, function symbols, etc. can be used; usually B is the formula million part of it. Distinguish In (1) ... ... Great Polytechnic Encyclopedia

We can write some mathematical expressions in different ways. Depending on our goals, whether we have enough data, etc. Numeric and Algebraic Expressions differ in that we write the first only as numbers combined with the help of signs of arithmetic operations (addition, subtraction, multiplication, division) and brackets.

If instead of numbers you enter Latin letters (variables) into the expression, it will become algebraic. Algebraic expressions use letters, numbers, signs of addition and subtraction, multiplication and division. And also the sign of the root, degree, brackets can be used.

In any case, whether this expression is numerical or algebraic, it cannot be just a random set of characters, numbers and letters - it must have a meaning. This means that letters, numbers, signs must be connected by some kind of relationship. Correct example: 7x + 2: (y + 1). Bad example): + 7x - * 1.

The word "variable" was mentioned above - what does it mean? This is a Latin letter, instead of which you can substitute a number. And if we are talking about variables, in this case, algebraic expressions can be called an algebraic function.

The variable can take on different values. And substituting some number in its place, we can find the value of the algebraic expression for this particular value of the variable. When the value of the variable is different, the value of the expression will also be different.

How to solve algebraic expressions?

To calculate the values you need to do transformation of algebraic expressions. And for this you still need to consider a few rules.

First: the domain of an algebraic expression is all possible values of a variable for which the expression can make sense. What is meant? For example, you cannot substitute a value for a variable that would require you to divide by zero. In the expression 1 / (x - 2), 2 must be excluded from the domain of definition.

Secondly, remember how to simplify expressions: factorize, bracket identical variables, etc. For example: if you swap the terms, the sum will not change (y + x = x + y). Similarly, the product will not change if the factors are interchanged (x * y \u003d y * x).

In general, they are excellent for simplifying algebraic expressions. abbreviated multiplication formulas. Those who have not yet learned them should definitely do this - they will still come in handy more than once:

we find the difference of the variables squared: x 2 - y 2 \u003d (x - y) (x + y);

we find the sum squared: (x + y) 2 \u003d x 2 + 2xy + y 2;

we calculate the difference squared: (x - y) 2 \u003d x 2 - 2xy + y 2;

we cube the sum: (x + y) 3 \u003d x 3 + 3x 2 y + 3xy 2 + y 3 or (x + y) 3 \u003d x 3 + y 3 + 3xy (x + y);

cube the difference: (x - y) 3 \u003d x 3 - 3x 2 y + 3xy 2 - y 3 or (x - y) 3 \u003d x 3 - y 3 - 3xy (x - y);

we find the sum of the variables cubed: x 3 + y 3 \u003d (x + y) (x 2 - xy + y 2);

we calculate the difference of the variables cubed: x 3 - y 3 \u003d (x - y) (x 2 + xy + y 2);

we use the roots: xa 2 + ya + z \u003d x (a - a 1) (a - a 2), and 1 and a 2 are the roots of the expression xa 2 + ya + z.

You should also have an idea about the types of algebraic expressions. They are:

rational, and those in turn are divided into:

integers (they do not have division into variables, there is no extraction of roots from variables and there is no raising to a fractional power): 3a 3 b + 4a 2 b * (a - b). The scope is all possible values of variables;

fractional (except for other mathematical operations, such as addition, subtraction, multiplication, in these expressions they divide by a variable and raise to a power (with a natural exponent): (2 / b - 3 / a + c / 4) 2. Domain of definition - all values variables for which the expression is not equal to zero;

irrational - in order for an algebraic expression to be considered as such, it must contain the exponentiation of variables to a power with a fractional exponent and / or the extraction of roots from variables: √a + b 3/4. The domain of definition is all values of the variables, excluding those in which the expression under the root of an even degree or under a fractional degree becomes a negative number.

Identity transformations of algebraic expressions is another useful technique for solving them. An identity is an expression that will be true for any variables included in the domain of definition that are substituted into it.

An expression that depends on some variables can be identically equal to another expression if it depends on the same variables and if the values of both expressions are equal, whichever values of the variables are chosen. In other words, if an expression can be expressed in two different ways (expressions) whose values are the same, these expressions are identically equal. For example: y + y \u003d 2y, or x 7 \u003d x 4 * x 3, or x + y + z \u003d z + x + y.

When performing tasks with algebraic expressions, the identical transformation serves to ensure that one expression can be replaced by another, identical to it. For example, replace x 9 with the product x 5 * x 4.

Solution examples

To make it clearer, let's look at a few examples. transformations of algebraic expressions. Tasks of this level can be found in KIMs for the Unified State Examination.

Task 1: Find the value of the expression ((12x) 2 - 12x) / (12x 2 -1).

Solution: ((12x) 2 - 12x) / (12x 2 - 1) \u003d (12x (12x -1)) / x * (12x - 1) \u003d 12.

Task 2: Find the value of the expression (4x 2 - 9) * (1 / (2x - 3) - 1 / (2x +3).

Solution: (4x 2 - 9) * (1 / (2x - 3) - 1 / (2x + 3) \u003d (2x - 3) (2x + 3) (2x + 3 - 2x + 3) / (2x - 3 )(2x + 3) = 6.

Conclusion

When preparing for school tests, USE and GIA exams, you can always use this material as a hint. Keep in mind that an algebraic expression is a combination of numbers and variables expressed in Latin letters. And also signs of arithmetic operations (addition, subtraction, multiplication, division), brackets, degrees, roots.

Use short multiplication formulas and your knowledge of identities to transform algebraic expressions.

Write us your comments and wishes in the comments - it is important for us to know that you are reading us.

site, with full or partial copying of the material, a link to the source is required.

The publication presents the logic of differences in algebraic expressions for students of basic general and secondary (complete) general education as a transitional stage in the formation of the logic of differences in mathematical expressions used in physics, etc. for the formation in the future of concepts about phenomena, tasks, their classification and methodology of the approach to their solution.

Download:

Preview:

Algebraic expressions and their characteristics

Algebra, as a science, studies the patterns of actions on sets, denoted by letters.Algebraic operations include addition, subtraction, multiplication, division, exponentiation, and root extraction.As a result of these actions, algebraic expressions were formed.Algebraic expression - an expression consisting of numbers and letters denoting sets, with which algebraic operations are performed.These actions passed into algebra from arithmetic. In algebra, one considersequating one algebraic expression to another, which is their identical equality. Examples of algebraic expressions are given in §1.Methods of transformations and relationships of expressions were also borrowed from arithmetic. Knowledge of the arithmetic patterns of actions on arithmetic expressions allows you to perform transformations on similar algebraic expressions, transform them, simplify, compare, analyze.Algebra is the science of regularities of transformations of expressions, consisting of sets presented in the form of letter designations, interconnected by signs of various actions.There are also more complex algebraic expressions studied in higher educational institutions. While they can be divided into types, the most commonly used in the school course.

1 Types of algebraic expressions

item 1 Simple expressions: 4a; (a+b); (a + b)3c; ; .

item 2 Identity equalities:(a + b)c = ac + bc; ;

item 3 Inequalities: as ; a + c .

p.4 Formulas: x=2a+5; y=3b; y \u003d 0.5d 2 +2;

p.5 Proportions:

First level of difficulty

Second level of difficulty

Third level of difficultyin terms of finding values for sets

a, b, c, m, k, d:

Fourth level of difficultyfrom the point of view of searching for values for sets a, y:

p.6 Equations:

ax + c \u003d -5bx; 4x 2 + 2x = 42;

Etc.

item 7 Functional dependencies: y=3x; y=ax 2 +4b; y \u003d 0.5x 2 +2;

Etc.

2 Consider algebraic expressions

2.1 Section 1 presents simple algebraic expressions. There is a view and

more difficult, for example:

As a rule, such expressions do not have the "=" sign. The task when considering such expressions is to transform them and obtain them in a simplified form. When converting the algebraic expression related to claim 1, a new algebraic expression is obtained, which is equivalent in meaning to the previous one. Such expressions are said to be identically equivalent. Those. the algebraic expression to the left of the equals sign is equivalent in its meaning to the algebraic expression to the right. In this case, an algebraic expression of a new kind is obtained, called the identical equality (see item 2).

2.2 Section 2 presents the algebraic identity equalities, which are formed with algebraic methods of transformation, algebraic expressions are considered, which are most often used as methods in solving problems in physics. Examples of identical equalities of algebraic transformations that are often used in mathematics and physics:

Commutative law of addition: a + b = b + a.

Associative law of addition:(a + b) + c = a + (b + c).

Commutative law of multiplication: ab=ba.

Associative law of multiplication:(ab)c = a(bc).

The distributive law of multiplication with respect to addition:

(a + b)c = ac + bc.

The distributive law of multiplication with respect to subtraction:

(a - b)c \u003d ac - bc.

Identity equalitiesfractional algebraic expressions(it is assumed that the denominators of the fractions are non-zero):

Identity equalitiesalgebraic expressions with powers:

A) ,

where (n times, ) - degree with integer exponent

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

Identity equalitiesalgebraic expressions with roots nth degree:

Expression - arithmetic root n th degree from among In particular, - arithmetic square.

Degree with a fractional (rational) exponent root:

The equivalent equivalent expressions given above are used to transform more complex algebraic expressions that do not contain the “=” sign.

Let us consider an example in which, for transformations of a more complex algebraic expression, the knowledge acquired during the transformations of simpler algebraic expressions in the form of identical equalities is used.

2.3 Section 3 presents the algebraic equality, for which the algebraic expression of the left side is not equal to the right side, i.e. are not identical. In this case, they are inequalities. As a rule, when solving some problems in physics, the properties of inequalities are important:

1) If a , then for any c : a + c .

2) If a and c > 0, then as .

3) If a and c , then ac > bc .

4) If a , a and b one sign, then 1/a > 1/b .

5) If a and c , then a + c , a - d .

6) If a , c , a > 0 , b > 0 , c > 0 , d > 0 , then ac .

7) If a , a > 0 , b > 0 , then

8) If , then

2.4 Section 4 presents the algebraic formulasthose. algebraic expressions that have a letter on the left side of the equal sign, denoting a set whose value is unknown and must be determined. And on the right side of the equal sign there are sets whose values are known. In this case, this algebraic expression is called an algebraic formula.

An algebraic formula is an algebraic expression containing an equal sign, on the left side of which there is a set whose value is unknown, and on the right side there are sets with known values, based on the condition of the problem.To determine the unknown value of the set to the left of the "equals" sign, the known values of the quantities on the right side of the "equals" sign are substituted and the arithmetic computational operations indicated in the algebraic expression in this part are performed.

Example 1:

Given: Solution:

a=25 Let the algebraic expression be given:

x=? x=2a+5.

This algebraic expression is an algebraic formula since to the left of the equals sign is the set whose value is to be found, and to the right are the sets with known values.

Therefore, it is possible to perform the substitution of the known value for the set "a", to determine the unknown value of the set "x":

x=2 25+5=55. Answer: x=55.

Example 2:

Given: Solution:

a=25 Algebraic expressionis a formula.

b=4 Therefore, it is possible to perform substitution of known

c=8 values for sets to the right of the equals sign,

d=3 to determine the unknown value of the set "k",

m=20 standing on the left:

n=6 Answer: k=3.2.

QUESTIONS

1 What is an algebraic expression?

2 What kinds of algebraic expressions do you know?

3 What algebraic expression is called identical equality?

4 Why is it necessary to know the patterns of identical equalities?

5 What algebraic expression is called a formula?

6 What algebraic expression is called an equation?

7 What algebraic expression is called functional dependence?

Numeric and algebraic expressions. Expression conversion.

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

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's happened 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. This 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's happened 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, it happens, sometimes, 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- It's 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 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?

Certainly. 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? Certainly. 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. This 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.)

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.