Conversions of irrational expressions

The article reveals the meaning of rational expressions and transformations with them. Let's consider the very concept of irrational expressions, transformation and characteristic expressions.

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What are irrational expressions?

When introducing roots at school, we study the concept of irrational expressions. Such expressions are closely related to roots.

Definition 1

Irrational expressions are expressions that have a root. That is, these are expressions that have radicals.

Based on this definition, we have that x - 1 , 8 3 3 6 - 1 2 3 , 7 - 4 3 (2 + 3) , 4 a 2 d 5: d 9 2 a 3 5 - these are all expressions of an irrational type.

When considering the expression x · x - 7 · x + 7 x + 3 2 · x - 8 3 we find that the expression is rational. Rational expressions include polynomials and algebraic fractions. Irrational ones include working with logarithmic expressions or radical expressions.

Main types of transformations of irrational expressions

When calculating such expressions, it is necessary to pay attention to the DZ. Often they require additional transformations in the form of opening parentheses, bringing similar members, groupings, and so on. The basis of such transformations is operations with numbers. Transformations of irrational expressions adhere to a strict order.

Example 1

Transform the expression 9 + 3 3 - 2 + 4 · 3 3 + 1 - 2 · 3 3 .

Solution

It is necessary to replace the number 9 with an expression containing the root. Then we get that

81 + 3 3 - 2 + 4 3 3 + 1 - 2 3 3 = = 9 + 3 3 - 2 + 4 3 3 + 1 - 2 3 3

The resulting expression has similar terms, so let's perform the reduction and grouping. We get

9 + 3 3 - 2 + 4 3 3 + 1 - 2 3 3 = = 9 - 2 + 1 + 3 3 + 4 3 3 - 2 3 3 = = 8 + 3 3 3
Answer: 9 + 3 3 - 2 + 4 3 3 + 1 - 2 3 3 = 8 + 3 3 3

Example 2

Present the expression x + 3 5 2 - 2 · x + 3 5 + 1 - 9 as a product of two irrationals using abbreviated multiplication formulas.

Solutions

x + 3 5 2 - 2 x + 3 5 + 1 - 9 = = x + 3 5 - 1 2 - 9

We represent 9 in the form of 3 2, and we apply the formula for the difference of squares:

x + 3 5 - 1 2 - 9 = x + 3 5 - 1 2 - 3 2 = = x + 3 5 - 1 - 3 x + 3 5 - 1 + 3 = = x + 3 5 - 4 x + 3 5 + 2

The result of identical transformations led to the product of two rational expressions that needed to be found.

Answer:

x + 3 5 2 - 2 x + 3 5 + 1 - 9 = = x + 3 5 - 4 x + 3 5 + 2

You can perform a number of other transformations that apply to irrational expressions.

Converting a Radical Expression

The important thing is that the expression under the root sign can be replaced by one that is identically equal to it. This statement makes it possible to work with a radical expression. For example, 1 + 6 can be replaced by 7 or 2 · a 5 4 - 6 by 2 · a 4 · a 4 - 6 . They are identically equal, so the replacement makes sense.

When there is no a 1 different from a, where an inequality of the form a n = a 1 n is valid, then such an equality is possible only for a = a 1. The values ​​of such expressions are equal to any values ​​of the variables.

Using Root Properties

The properties of roots are used to simplify expressions. To apply the property a · b = a · b, where a ≥ 0, b ≥ 0, then from the irrational form 1 + 3 · 12 can become identically equal to 1 + 3 · 12. Property. . . a n k n 2 n 1 = a n 1 · n 2 · , . . . , · n k , where a ≥ 0 means that x 2 + 4 4 3 can be written in the form x 2 + 4 24 .

There are some nuances when converting radical expressions. If there is an expression, then - 7 - 81 4 = - 7 4 - 81 4 we cannot write it down, since the formula a b n = a n b n serves only for non-negative a and positive b. If the property is applied correctly, then the result will be an expression of the form 7 4 81 4 .

For correct transformation, transformations of irrational expressions using the properties of roots are used.

Entering a multiplier under the sign of the root

Place under the root sign– means to replace the expression B · C n , and B and C are some numbers or expressions, where n is natural number, which is greater than 1, is equal to an expression that looks like B n · C n or - B n · C n .

If we simplify the expression of the form 2 x 3, then after adding it to the root, we get that 2 3 x 3. Such transformations are possible only after a detailed study of the rules for introducing a multiplier under the root sign.

Removing the multiplier from under the root sign

If there is an expression of the form B n · C n , then it is reduced to the form B · C n , where there are odd n , which take the form B · C n with even n , B and C being some numbers and expressions.

That is, if we take an irrational expression of the form 2 3 x 3, remove the factor from under the root, then we get the expression 2 x 3. Or x + 1 2 · 7 will result in an expression of the form x + 1 · 7, which has another notation of the form x + 1 · 7.

Removing the multiplier from under the root is necessary to simplify the expression and quickly convert it.

Converting fractions containing roots

An irrational expression can be either a natural number or a fraction. To convert fractional expressions, pay great attention to its denominator. If we take a fraction of the form (2 + 3) x 4 x 2 + 5 3, then the numerator will take the form 5 x 4, and, using the properties of the roots, we find that the denominator will become x 2 + 5 6. The original fraction can be written as 5 x 4 x 2 + 5 6.

It is necessary to pay attention to the fact that it is necessary to change the sign of only the numerator or only the denominator. We get that

X + 2 x - 3 x 2 + 7 4 = x + 2 x - (- 3 x 2 + 7 4) = x + 2 x 3 x 2 - 7 4

Reducing a fraction is most often used when simplifying. We get that

3 · x + 4 3 - 1 · x x + 4 3 - 1 3 reduce by x + 4 3 - 1 . We get the expression 3 x x + 4 3 - 1 2.

Before reduction, it is necessary to perform transformations that simplify the expression and make it possible to factorize complex expression. Abbreviated multiplication formulas are most often used.

If we take a fraction of the form 2 · x - y x + y, then it is necessary to introduce new variables u = x and v = x, then the given expression will change form and become 2 · u 2 - v 2 u + v. The numerator should be decomposed into polynomials according to the formula, then we get that

2 · u 2 - v 2 u + v = 2 · (u - v) · u + v u + v = 2 · u - v . After performing the reverse substitution, we arrive at the form 2 x - y, which is equal to the original one.

Reduction to a new denominator is allowed, then it is necessary to multiply the numerator by an additional factor. If we take a fraction of the form x 3 - 1 0, 5 · x, then we reduce it to the denominator x. to do this, you need to multiply the numerator and denominator by the expression 2 x, then we get the expression x 3 - 1 0, 5 x = 2 x x x 3 - 1 0, 5 x 2 x = 2 x x 3 - 1 x .

Reducing fractions or bringing similar ones is necessary only on the ODZ of the specified fraction. When we multiply the numerator and denominator by an irrational expression, we find that we get rid of the irrationality in the denominator.

Getting rid of irrationality in the denominator

When an expression gets rid of the root in the denominator by transformation, it is called getting rid of irrationality. Let's look at the example of a fraction of the form x 3 3. After getting rid of irrationality, we obtain a new fraction of the form 9 3 x 3.

Transition from roots to powers

Transitions from roots to powers are necessary for quickly transforming irrational expressions. If we consider the equality a m n = a m n , we can see that its use is possible when a is a positive number, m is an integer, and n is a natural number. If we consider the expression 5 - 2 3, then otherwise we have the right to write it as 5 - 2 3. These expressions are equivalent.

When the root has a negative number or a number with variables, then the formula a m n = a m n is not always applicable. If you need to replace such roots (- 8) 3 5 and (- 16) 2 4 with powers, then we get that - 8 3 5 and - 16 2 4 by the formula a m n = a m n we do not work with negative a. In order to analyze in detail the topic of radical expressions and their simplifications, it is necessary to study the article on the transition from roots to powers and back. It should be remembered that the formula a m n = a m n is not applicable to all expressions of this type. Getting rid of irrationality contributes to further simplification of the expression, its transformation and solution.

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Expressions containing a radical sign (root) are called irrational.

An arithmetic root of a natural power $n$ of a non-negative number a is some non-negative number such that when raised to the power $n$ the number $a$ is obtained.

$(√^n(a))^n=a$

In the notation $√^n(a)$, “a” is called the radical number, $n$ is the exponent of the root or radical.

Properties of $n$th roots for $a≥0$ and $b≥0$:

1. The root of the product is equal to the product of the roots

$√^n(a∙b)=√^n(a)∙√^n(b)$

Calculate $√^5(5)∙√^5(625)$

The root of a product is equal to the product of roots and vice versa: the product of roots with the same root exponent is equal to the root of the product of radical expressions

$√^n(a)∙√^n(b)=√^n(a∙b)$

$√^5{5}∙√^5{625}=√^5{5∙625}=√^5{5∙5^4}=√^5{5^5}=5$

2. The root of a fraction is a separate root from the numerator and a separate root from the denominator

$√^n((a)/(b))=(√^n(a))/(√^n(b))$, for $b≠0$

3. When a root is raised to a power, the radical expression is raised to this power

$(√^n(a))^k=√^n(a^k)$

4. If $a≥0$ and $n,k$ are natural numbers greater than $1$, then the equality is true.

$√^n(√^k(a))=√^(n∙k)a$

5. If the indicators of the root and radical expression are multiplied or divided by the same natural number, then the value of the root will not change.

$√^(n∙m)a^(k∙m)=√^n(a^k)$

6. The root of an odd degree can be extracted from positive and negative numbers, and the root of an even degree - only from positive ones.

7. Any root can be represented as a power with a fractional (rational) exponent.

$√^n(a^k)=a^((k)/(n))$

Find the value of the expression $(√(9∙√^11(s)))/(√^11(2048∙√s))$ for $s>0$

The root of the product is equal to the product of the roots

$(√(9∙√^11(s)))/(√^11(2048∙√s))=(√9∙√(√^11(s)))/(√^11(2048)∙ √^11(√с))$

We can extract roots from numbers immediately

$(√9∙√(√^11(s)))/(√^11(2048)∙√^11(√s))=(3∙√(√^11(s)))/(2∙ √^11(√с))$

$√^n(√^k(a))=√^(n∙k)a$

$(3∙√(√^11(s)))/(2∙√^11(√s))=(3∙√^22(s))/(2∙√^22(s))$

We reduce the $22$ roots of $с$ and get $(3)/(2)=1.5$

Answer: $1.5$

If for a radical with an even exponent we do not know the sign of the radical expression, then when extracting the root, the module of the radical expression comes out.

Find the value of the expression $√((с-7)^2)+√((с-9)^2)$ at $7< c < 9$

If there is no exponent above the root, this means that we are working with a square root. Its indicator is two, i.e. honest. If for a radical with an even exponent we do not know the sign of the radical expression, then when extracting the root, the module of the radical expression comes out.

$√((с-7)^2)+√((с-9)^2)=|c-7|+|c-9|$

Let's determine the sign of the expression under the modulus sign based on the condition $7< c < 9$

To check, take any number from a given range, for example, $8$

Let's check the sign of each module

$8-9<0$, при раскрытии модуля пользуемся правилом: модуль положительного числа равен самому себе, отрицательного числа - равен противоположному значению. Так как у второго модуля знак отрицательный, при раскрытии меняем знак перед модулем на противоположный.

$|c-7|+|c-9|=(с-7)-(с-9)=с-7-с+9=2$

Properties of powers with rational exponent:

1. When multiplying powers with the same bases, the base remains the same, and the exponents are added.

$a^n∙a^m=a^(n+m)$

2. When raising a degree to a power, the base remains the same, but the exponents are multiplied

$(a^n)^m=a^(n∙m)$

3. When raising a product to a power, each factor is raised to this power

$(a∙b)^n=a^n∙b^n$

4. When raising a fraction to a power, the numerator and denominator are raised to this power

Identical transformations of expressions are one of the content lines of the school mathematics course. Identical transformations are widely used in solving equations, inequalities, systems of equations and inequalities. In addition, identical transformations of expressions contribute to the development of intelligence, flexibility and rationality of thinking.

The proposed materials are intended for 8th grade students and include the theoretical foundations of identical transformations of rational and irrational expressions, types of tasks for transforming such expressions and the text of the test.

1. Theoretical foundations of identity transformations

Expressions in algebra are records consisting of numbers and letters connected by action signs.

https://pandia.ru/text/80/197/images/image002_92.gif" width="77" height="21 src=">.gif" width="20" height="21 src="> – algebraic expressions.

Depending on the operations, rational and irrational expressions are distinguished.

Algebraic expressions are called rational if relative to the letters included in it A, b, With, ... no other operations are performed except addition, multiplication, subtraction, division and exponentiation.

Algebraic expressions containing operations of extracting the root of a variable or raising a variable to a rational power that is not an integer are called irrational with respect to this variable.

An identity transformation of a given expression is the replacement of one expression with another that is identically equal to it on a certain set.

The following theoretical facts underlie identical transformations of rational and irrational expressions.

1. Properties of degrees with an integer exponent:

, n ON; A 1=A;

, n ON, A¹0; A 0=1, A¹0;

, A¹0;

, A¹0;

, A¹0;

, A¹0, b¹0;

, A¹0, b¹0.

2. Abbreviated multiplication formulas:

Where A, b, With– any real numbers;

Where A¹0, X 1 and X 2 – roots of the equation .

3. The main property of fractions and actions on fractions:

, Where b¹0, With¹0;

; ;

4. Definition of an arithmetic root and its properties:

; , b#0; https://pandia.ru/text/80/197/images/image026_24.gif" width="84" height="32">; ; ,

Where A, b– non-negative numbers, n ON, n³2, m ON, m³2.

1. Types of Expression Conversion Exercises

There are various types of exercises on identity transformations of expressions. First type: The conversion that needs to be performed is explicitly specified.

For example.

1. Represent it as a polynomial.

When performing this transformation, we used the rules of multiplication and subtraction of polynomials, the formula for abbreviated multiplication and the reduction of similar terms.

2. Factor into: .

When performing the transformation, we used the rule of placing the common factor out of brackets and 2 abbreviated multiplication formulas.

3. Reduce the fraction:

.

When performing the transformation, we used the removal of the common factor from brackets, commutative and contractile laws, 2 abbreviated multiplication formulas, and operations on powers.

4. Remove the factor from under the root sign if A³0, b³0, With³0: https://pandia.ru/text/80/197/images/image036_17.gif" width="432" height="27">

We used the rules for actions on roots and the definition of the modulus of a number.

5. Eliminate irrationality in the denominator of a fraction. .

Second type exercises are exercises in which the main transformation that needs to be performed is clearly indicated. In such exercises, the requirement is usually formulated in one of the following forms: simplify the expression, calculate. When performing such exercises, it is necessary first of all to identify which and in what order transformations need to be performed so that the expression takes on a more compact form than the given one, or a numerical result is obtained.

For example

6. Simplify the expression:

Solution:

.

Used action rules algebraic fractions and abbreviated multiplication formulas.

7. Simplify the expression:

.

If A³0, b³0, A¹ b.

We used abbreviated multiplication formulas, rules for adding fractions and multiplying irrational expressions, the identity https://pandia.ru/text/80/197/images/image049_15.gif" width="203" height="29">.

We used the operation of selecting a complete square, the identity https://pandia.ru/text/80/197/images/image053_11.gif" width="132 height=21" height="21">, if .

Proof:

Since , then and or or or , i.e. .

We used the condition and formula for the sum of cubes.

It should be borne in mind that conditions connecting variables can also be specified in exercises of the first two types.

For example.

10. Find if .

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Trainer No. 1

Topic: Converting power and irrational expressions

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