(1) a m ⋅ a n \u003d a m + n

Example:

$$ (a ^ 2) \\ Cdot (a ^ 5) \u003d (a ^ 7) $$ (2) a m a n \u003d a m - n

Example:

$$ \\ FRAC (((a ^ 4))) (((a ^ 3))) \u003d (a ^ (4 - 3)) \u003d (a ^ 1) \u003d a $$ (3) (a ⋅ b) n \u003d an ⋅ bn

Example:

$$ ((A \\ Cdot b) ^ 3) \u003d (a ^ 3) \\ Cdot (B ^ 3) $$ (4) (a b) n \u003d a n b n

Example:

$$ (\\ left ((\\ FRAC (A) (B)) \\ RIGHT) ^ 8) \u003d \\ FRAC (((A ^ 8))) (((b ^ 8))) $$ (5) (am ) n \u003d am ⋅ n

Example:

$$ (((a ^ 2)) ^ 5) \u003d (a ^ (2 \\ Cdot 5)) \u003d (a ^ (10)) $$ (6) a - n \u003d 1 a n

Examples:

$$ (a ^ (- 2)) \u003d \\ FRAC (1) (((a ^ 2))); \\; \\; \\; \\; (a ^ (- 1)) \u003d \\ FRAC (1) (( (A ^ 1))) \u003d \\ FRAC (1) (a). $$

Properties square root:

(1) a b \u003d a ⋅ b, at a ≥ 0, b ≥ 0

Example:

18 = 9 ⋅ 2 = 9 ⋅ 2 = 3 2

(2) a b \u003d a b, at a ≥ 0, b\u003e 0

Example:

4 81 = 4 81 = 2 9

(3) (a) 2 \u003d A, with a ≥ 0

Example:

(4) a 2 \u003d | a | With any A.

Examples:

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

Rational and irrational numbers

Rational numbers - numbers that can be represented as ordinary fraci M n where m is an integer (ℤ \u003d 0, ± 1, ± 2, ± 3 ...), n - natural (ℕ \u003d 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 \u003d 2,71828182845 ...

π \u003d 3,1415926 ...

2 = 1,414213562…

3 = 1,7320508075…

Simply put, irrational numbers are numbers containing square root sign in their record. But not everything is so simple. Some rational numbers are masked under irrational, for example, the number 4 contains a square root sign in its record, but we perfectly understand that you can simplify the form of recording 4 \u003d 2. This means that the number 4 is the number is rational.

Similarly, the number 4 81 \u003d 4 81 \u003d 2 9 is the number rational.

In some tasks, it is required to determine which of the numbers are rational, and which irrational. The task is reduced to understand what kind of irrational numbers, and which are disguised under them. To do this, you need to be able to perform a multiplier transaction operations from under the sign of a square root and making a multiplier under the sign of the root.

Multiplier making and making a sign of square root

With the help of making a multiplier for a square root sign, you can significantly simplify some mathematical expressions.

Example:

Simplify expression 2 8 2.

1 method (making a multiplier from under the root): 2 8 2 = 2 4 ⋅ 2 2 = 2 4 ⋅ 2 2 = 2 ⋅ 2 = 4

2 way (making a multiplier under the root sign): 2 8 2 = 2 2 8 2 = 4 ⋅ 8 2 = 4 ⋅ 8 2 = 16 = 4

Abbreviated multiplication formulas (FSA)

Square amount

(1) (A + B) 2 \u003d A 2 + 2 A B + B 2

Example:

(3 x + 4 y) 2 \u003d (3 x) 2 + 2 ⋅ 3 x ⋅ 4 y + (4 y) 2 \u003d 9 x 2 + 24 x y + 16 y 2

Square difference

(2) (a - b) 2 \u003d A 2 - 2 A B + B 2

Example:

(5 x - 2 y) 2 \u003d (5 x) 2 - 2 ⋅ 5 x ⋅ 2 y + (2 y) 2 \u003d 25 x 2 - 20 x y + 4 y 2

The sum of the squares does not decline for multipliers

Square differences

(3) a 2 - b 2 \u003d (a - b) (a + b)

Example:

25 x 2 - 4 y 2 \u003d (5 x) 2 - (2 y) 2 \u003d (5 x - 2 y) (5 x + 2 y)

Cube amount

(4) (A + B) 3 \u003d A 3 + 3 A 2 B + 3 A B 2 + B 3

Example:

(x + 3 y) 3 \u003d (x) 3 + 3 ⋅ (x) 2 ⋅ (3 y) + 3 ⋅ (x) ⋅ (3 y) 2 + (3 y) 3 \u003d x 3 + 3 ⋅ x 2 ⋅ 3 y + 3 ⋅ x ⋅ 9 y 2 + 27 y 3 \u003d x 3 + 9 x 2 y + 27 xy 2 + 27 y 3

Cube difference

(5) (A - B) 3 \u003d A 3 - 3 A 2 B + 3 A B 2 - B 3

Example:

(x 2 - 2 y) 3 \u003d (x 2) 3 - 3 ⋅ (x 2) 2 ⋅ (2 y) + 3 ⋅ (x 2) ⋅ (2 y) 2 - (2 y) 3 \u003d x 2 3 - 3 ⋅ x 2 ⋅ 2 ⋅ 2 y + 3 ⋅ x 2 ⋅ 4 y 2 - 8 y 3 \u003d x 6 - 6 x 4 y + 12 x 2 y 2 - 8 y 3

The amount of cubes

(6) A 3 + B 3 \u003d (A + B) (A 2 - A B + B 2)

Example:

8 + x 3 \u003d 2 3 + x 3 \u003d (2 + x) (2 2 - 2 ⋅ x + x 2) \u003d (x + 2) (4 - 2 x + x 2)

Cubic differences

(7) a 3 - b 3 \u003d (a - b) (A 2 + A B + B 2)

Example:

x 6 - 27 y 3 \u003d (x 2) 3 - (3 y) 3 \u003d (x 2 - 3 y) ((x 2) 2 + (x 2) (3 y) + (3 y) 2) \u003d ( x 2 - 3 y) (x 4 + 3 x 2 y + 9 y 2)

Standard view of the number

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

First digit number They call it the first left different from zero number.

Examples:
2 5; 3, 05; 0, 1 43; 0, 00 1 2. The first meaning digit is highlighted in red.

In order to bring the number to the standard form, it is necessary:

1. Move the comma so that it is immediately behind the first meaning digit.
2. The resulting number is multiplied by 10 n, where n is a number that is defined as follows:
3. n\u003e 0 if the comma shifted to the left (multiplication of 10 n, indicates that actually the comma should be to the right);
4. n.< 0 , если запятая сдвигалась вправо (умножение на 10 n , указывает, что на самом деле запятая должна стоять левее);
5. the absolute value of the number n is equal to the number of discharges to which the comma was shifted.

Examples:

25 = 2 , 5 ← ​ , = 2,5 ⋅ 10 1

The comma shifted to the left for 1 category. Since the shift shift is carried out to the left, the degree is positive.

Already given to the standard form, do nothing with it. It can be written as 3.05 ⋅ 10 0, but since 10 0 \u003d 1, we leave the number in its original form.

0,143 = 0, 1 → , 43 = 1,43 ⋅ 10 − 1

The comma shifted to the right on 1 category. Since the shift shift is right, the degree is negative.

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

The comma shifted to the right of three discharge. Since the shift shift is right, the degree is negative.

Algebraic expression

the expression composed of letters and numbers connected by the signs of action of addition, subtraction, multiplication, divisions, the construction of the root and the extraction of the root (degree and root indicators should be permanent). A. in. called rational relative to some letters, in it incoming if it does not contain them under the sign of the root extraction, for example

rationally relative to A, B and C. A. in. It is called as much regarding some letters if it does not contain divisions on the expressions containing these letters, for example 3A / C + BC 2 - 3As / 4 it is only relative to A and B. If some of the letters (or all) consider variables, then A. in. There is an algebraic function.

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

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Algebraic expressions are made up of numbers and variables using the signs of addition, subtraction, multiplication, divisions, erection in the rational degree and extraction of the root and with the help of brackets.

Consider some examples of algebraic expressions:

2A 2 B - 3ab 2 (A + B)

(1 / A + 1 / B - C / 3) 3.

There are several types of algebraic expressions.

It is called such an algebraic expression that does not contain divisions on variables and the extraction of the root from variables (including the construction of a fractional indicator).

2A 2 B - 3ab 2 (A + B) is an algebraic expression.

(1 / a + 1 / b - C / 3) 3 is not an algebraic expression, because Contains division into a variable.

The fractional is called such an algebraic expression, which is composed of among the numbers and variables through the actions of addition, subtraction, multiplication, erection in a ratio with a natural indicator and division.

(1 / A + 1 / B - C / 3) 3 is a fractional algebraic expression.

Rational algebraic expressions are called entire and fractional expressions.

It means that 2a 2 b - 3ab 2 (a + b), and (1 / a + 1 / b - C / 3) 3 are rational algebraic expressions.

An irrational algebraic expression is such an algebraic expression that uses the root extraction from variables (or the construction of variables into a fractional degree).

a 2/3 - B 2/3 - an irrational algebraic expression.

In other words, all algebraic expressions are divided into two large groups: rational and irrational algebraic expressions. Rational expressionsIn turn, are divided into whole and fractional.

The permissible value of variables is called such a value of the variables in which the algebraic expression makes sense. The set of all permissible values \u200b\u200bof the variable is the area of \u200b\u200bdefinition of an algebraic expression.

Extent expressions make sense with any values \u200b\u200bof its variables. For example, 2A 2 B - 3ab 2 (A + B) makes sense and at a \u003d 0, b \u003d 1, and at a \u003d 3, b \u003d 6, etc.

Suppose that a \u003d 0, b \u003d 1, and let's try to find an expression solution

2A 2 B - 3ab 2 (A + B).

If a \u003d 0, b \u003d 1, then 2 ∙ 0 2 ∙ 1 - 3 ∙ 0 ∙ 1 2 ∙ (0 + 1) \u003d 0 ∙ 0 \u003d 0.

So, at a \u003d 0, b \u003d 1, the expression is 0.

Fractional expressions make sense only if the values \u200b\u200bdo not pay variables into zero: we can not be divided into zero to zero.

The expression (1 / a + 1 / b - C / 3) 3 makes sense at a and b not equal zero (a ≠ 0, b ≠ 0). Otherwise, we will receive division to zero.

The irrational expression will not make sense with the values \u200b\u200bof the variables that pay a negative number of expression contained under the rope of an even degree or under the sign of the construction rate.

The expression A 2/3 - B 2/3 makes sense at a ≥ 0 and b ≥ 0. Otherwise, we will encounter the construction of a negative number in the fractional degree.

The value of an algebraic expression is called a numerical expression resulting from the fact that the variables gave permissible values.

Find the value of an algebraic expression

a + B + C / 5 at a \u003d 6, b \u003d 3, c \u003d 5.

1. The expression A + B + C / 5 is a whole algebraic expression → all values \u200b\u200bare permissible.

2. Substitute the numeric values \u200b\u200bof variables and get:

6 + 3 + 5/5 = 9 + 1 = 10.

So, the answer is: 10.

The identity is called equality that is true with all the permissible values \u200b\u200bof the variables included in it.

They are identically equal to expressions, the corresponding values \u200b\u200bof which coincide with all permissible values \u200b\u200bof variables. So, the expressions x 5 and x 2 ∙ x 3, a + b + c and b + c + a are identical to each other.

The concept of identically equal expressions leads us to another important concept - identical conversion of expressions.

The identical transformation of the expression is called a replacement of one expression to another, identically equal to it.

This means that the expression X 5 can be identically converted to the expression x 2 ∙ x 3.

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Algebraic expression - This is any entry from the letters, numbers, arithmetic actions and brackets, drawn up with the meaning. In fact, the algebraic expression is a numerical expression in which the letters are also used in addition to numbers. Therefore, algebraic expressions are also called lettering expressions.

Basically, the letters of the Latin alphabet are used in alphabetic expressions. Why do these letters need? Instead, we can substitute different numbers. Therefore, these letters are called variables. That is, they can change their meaning.

Examples of algebraic expressions.

$ \\ begin (align) & x + 5; \\, \\, \\, \\, \\, (x + y) \\ centerdot (xy); \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\) (2) ; \\\\ \\\\ \\ \\ sqrt (((((b) ^ (2)) - 4ac); \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\ \\ \\ FRAC (1) (H); \\, \\, \\, \\, \\, a (((x) ^ (2)) + bx + c; \\\\\\ End (Align) $

If, for example, in the expression X + 5 we will substitute instead of a variable x some number, then we will obtain a numerical expression. At the same time, the value of this numerical expression will be the value of the algebraic expression X + 5 at a given value of the variable. That is, with x \u003d 10, x + 5 \u003d 10 + 5 \u003d 15. And at x \u003d 2, x + 5 \u003d 2 + 5 \u003d 7.

There are such values \u200b\u200bof the variable in which the algebraic expression loses its meaning. So, for example, it will be if in the expression 1: x we \u200b\u200bsubstitute instead of x value 0.
Since it is impossible to divide to zero.

The area of \u200b\u200bdefinition of an algebraic expression.

Many variable values \u200b\u200bin which the expression does not lose sense, called definition area This expression. It can also be said that the expression definition area is a set of all valid variable values.

Consider examples:

1. y + 5 - the definition area will be any values \u200b\u200bof Y.
2. 1: X - the expression will make sense at all x values \u200b\u200bexcept 0. Therefore, the definition area will be any X values \u200b\u200bexcept for zero.
3. (x + y) :( X-y) - the definition area is any values \u200b\u200bof X and Y, in which X ≠ Y.

Types of algebraic expressions.

Rational algebraic expressions - These are whole and fractional algebraic expressions.

1. A whole algebraic expression - does not contain the exercise to the degree with a fractional indicator, the extraction of the root from the variable, as well as divisions to the variable. In integral algebraic expressions, all values \u200b\u200bof variables are permissible. For example, AX + BX + C is an integer algebraic expression.
2. Fractional - contains division into a variable. $ \\ FRAC (1) (a) + BX + C $ - fractional algebraic expression. In fractional algebraic expressions, all values \u200b\u200bof variables are permissible, at which there is no division to zero.

Irrational algebraic expressions Contain root extraction from a variable or erection of a variable into a fractional degree.

$ \\ sqrt (((a) ^ (2)) + ((b) ^ (2))); \\, \\, \\, \\, \\, \\, \\, ((a) ^ (\\ FRAC (2) (3))) + ((b) ^ (\\ FRAC (1) (3))); $ - irrational algebraic expressions. In irrational algebraic expressions, all values \u200b\u200bof variables are permissible, under which the expression that stands under the rope is not negative.

Some mathematical expressions we can burn in different ways. Depending on our purposes, there is enough data to us, etc. Numeric and algebraic expressions They differ in the fact that the first we record only the numbers combined with the signs of arithmetic action (addition, subtraction, multiplication, division) and brackets.

If instead of the numbers introduce Latin letters (variables) to the expression, it will become algebraic. In algebraic expressions, letters, numbers, signs of addition and subtraction, multiplication and division are used. And the root, degree, brackets can also be used.

In any case, a numerical expression or algebraic, it cannot be just a random set of signs, numbers and letters - it should have a meaning. This means that letters, numbers, signs should be related to some kind of relationships. 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 various values. And substituting some number in its place, we can find the value of an algebraic expression at this particular value of the variable. When the value of the variable is odd, another will be the value of the expression.

How to solve algebraic expressions?

To calculate the values \u200b\u200byou need to do transformation of algebraic expressions. And for this you still need to take into account several rules.

First: the area of \u200b\u200bdefinition of algebraic expressions is all possible values \u200b\u200bof the variable in which this expression may make sense. What is meant? For example, it is impossible to substitute such a value of the variable at which it would have to be divided into zero. In expression1 / (x - 2) from the field of definition, it is necessary to exclude 2.

Secondly, remember how to simplify expressions: lay out the same variables for brackets, etc. For example: if we change the components, the amount from this will not change (u + x \u003d x + y). Similarly, the work will not change if there are multipliers (x * y \u003d y * x).

In general, to simplify algebraic expressions perfectly serve formulas of abbreviated multiplication. Those who have not learned them yet, be sure to do it - they will still use more than once:

we find the difference between the variables raised to the square: x 2 - in 2 \u003d (x - y) (x + y);

we find the amount erected into a square: (x + y) 2 \u003d x 2 + 2h + in 2;

calculate the difference erected into the square: (x - y) 2 \u003d x 2 - 2h + in 2;

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

cutting the difference: (x - y) 3 \u003d x 3 - 3x 2 y + 3h 2 - in 3 or (x - y) 3 \u003d x 3 - 3 - 3h (x - y);

we find the sum of the variables erected into the cubic: x 3 + in 3 \u003d (x + y) (x 2 - x + in 2);

calculate the difference between the variables raised to the cubic: x 3 - in 3 \u003d (x - y) (x 2 + x + in 2);

we use the roots: ha 2 + ua + z \u003d x (a - a 1) (a - a 2), and 1 and a 2 are the roots of the expression H 2 + UA + Z.

You still have an idea of \u200b\u200bthe types of algebraic expressions. They are:

rational, and those in turn are divided into:

whole (there is no division into variables in them, there is no root extraction from variables and there is no erection in a fractional degree): 3A 3 b + 4a 2 B * (a - b). The definition is all possible values \u200b\u200bof variables;

fractional (except for other mathematical operations, such as addition, subtraction, multiplication, in these expressions are divided into a variable and elevated to a degree (with a natural indicator): (2 / b - 3 / a + s / 4) 2. Definition area - all values variables in which the expression is not zero;

irrational - so that the algebraic expression was considered as such, it should present the construction of variables into a fractional indicator and / or extracting roots from the variables: √A + B 3/4. The definition area is all values \u200b\u200bof variables, excluding those in which the expression under the root of the even degree or under a fractional degree becomes a negative number.

Identical conversions of algebraic expressions - another useful reception To solve them. The parent is such an expression that will be faithful with any variables that are part of the definition of the variables that are subjected to 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 \u200b\u200bof both expressions are equal, what variable values \u200b\u200bare not selected. In other words, if the expression can be expressed in two different ways (expressions), the values \u200b\u200bof which are identical, these expressions are identically equal. For example: y + y \u003d 2a, or x 7 \u003d x 4 * x 3, or x + y + z \u003d z + x + y.

When performing tasks with algebraic expressions, the identical conversion is used so that one expression can be replaced by another identical to it. For example, replace x 9 on the product x 5 * x 4.

Examples of solutions

To be clearer, we will examine several examples transformation of algebraic expressions. The tasks of this level can get caught in Kima on the exam.

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) \u003d 6.

Conclusion

When preparing for school control, exams ege And GIA You can always use this material as a hint. Keep in memory that the algebraic expression is called a combination of numbers and variables expressed by Latin letters. And also signs arithmetic operations (addition, subtraction, multiplication, division), brackets, degrees, roots.

Use the formulas of abbreviated multiplication and knowledge of identical equalities to convert algebraic expressions.

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