Expressions containing a radical sign (root) are called irrational.
Arithmetic root natural degree$n$ from a non-negative number a is called some non-negative number, when raised to the power $n$, the number $a$ is obtained.
$(√^n(a))^n=a$
In the notation $√^n(a)$, “a” is called radical number, $n$ - an indicator of a root or radical.
Properties of $n$th roots for $a≥0$ and $b≥0$:
1. Root of the product equal to the product roots
$√^n(a∙b)=√^n(a)∙√^n(b)$
Calculate $√^5(5)∙√^5(625)$
The root of a product is equal to the product of the roots and vice versa: the product of the roots with the same indicator root 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. Root odd degree can be extracted from the positive and negative numbers, and the root of an even degree is only positive.
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 indicator above the root, this means that we are working with 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
The article reveals the meaning of irrational expressions and transformations with them. Let's consider the very concept of irrational expressions, transformation and characteristic expressions.
Yandex.RTB R-A-339285-1
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
Definition 3Place under the root sign- means to replace the expression B · C n, and B and C are some numbers or expressions, where n is a natural number that is greater than 1, with an equal 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 a 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 for 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 contains 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 for all expressions of this type. Getting rid of irrationality contributes to further simplification of the expression, its transformation and solution.
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The properties of the roots underlie the next two transformations, called bringing them under the root sign and taking them out from under the root sign, which we now consider.
Entering a multiplier under the sign of the root
Introducing a factor under the sign implies replacing the expression , where B and C are some numbers or expressions, and n is a natural number greater than one, with an identically equal expression of the form or .
For example, after introducing a factor of 2 under the root sign, an irrational expression takes the form .
The theoretical foundations of this transformation, the rules for its implementation, as well as solutions to various typical examples are given in the article introducing a multiplier under the sign of the root.
Removing the multiplier from under the root sign
A transformation, in a certain sense the opposite of introducing a factor under the root sign, is removing the factor from under the root sign. It consists of representing the root as a product for odd n or as a product for even n, where B and C are some numbers or expressions.
For an example, let’s return to the previous paragraph: the irrational expression, after removing the factor from under the root sign, takes the form . Another example: removing the factor from under the root sign in the expression gives the product, which can be rewritten as .
On what this transformation is based, and by what rules it is carried out, we will examine in a separate article the removal of the multiplier from under the sign of the root. There we will also give solutions to examples and list ways to reduce a radical expression to a form convenient for multiplying.
Converting fractions containing roots
Irrational expressions can contain fractions that have roots in the numerator and denominator. With such fractions you can carry out any of the basic identity transformations of fractions.
Firstly, nothing prevents you from working with expressions in the numerator and denominator. As an example, consider the fraction. The irrational expression in the numerator is obviously identically equal to , and by turning to the properties of roots, the expression in the denominator can be replaced by the root . As a result, the original fraction is converted to the form .
Second, you can change the sign in front of a fraction by changing the sign of the numerator or denominator. For example, the following transformations of an irrational expression take place: 
.
Thirdly, sometimes it is possible and advisable to reduce a fraction. For example, how to deny yourself the pleasure of reducing a fraction 
to the irrational expression, as a result we get 
.
It is clear that in many cases, before reducing a fraction, the expressions in its numerator and denominator have to be factored, which in simple cases can be achieved by abbreviated multiplication formulas. And sometimes it helps to reduce a fraction by replacing a variable, which allows you to move from the original fraction with irrationality to a rational fraction, which is more comfortable and familiar to work with.
For example, let's take the expression . Let's introduce new variables and , in these variables the original expression has the form . Having performed in the numerator
PRACTICAL WORK No. 1
Subject: "Transformation of algebraic, rational, irrational, power expressions."
Purpose of the work: learn to transform algebraic, rational, irrational, power expressions using abbreviated multiplication formulas, basic properties of roots and powers.
Theoretical information.
ROOTS OF NATURAL DEGREE FROM A NUMBER, THEIR PROPERTIES.
Root n – degrees : , n - root exponent, A - radical expression
If n – odd number, then the expression makes sense when A
If n – even number, then the expression makes sense when
Arithmetic root:
Odd root of a negative number:
BASIC PROPERTIES OF ROOTS
The rule for extracting the root from a product:
Rule for extracting a root from a root:
The rule for removing the multiplier from under the root sign:
Entering a multiplier under the root sign:
,
The index of the root and the index of the radical expression can be multiplied by the same number.
The rule for raising a root to a power.
DEGREE WITH NATURAL INDICATOR
= , a – the basis of the degree,n – exponent
Properties:
When multiplying powers with the same bases, the exponents are added, but the base remains unchanged.
When dividing degrees with the same bases, the exponents are subtracted, but the base remains unchanged.
When raising a power to a power, the exponents are multiplied.
When raising the product of two numbers to a power, each number is raised to that power and the results are multiplied.
If the quotient of two numbers is raised to a power, then the numerator and denominator are raised to this power, and the result is divided by each other.
DEGREE WITH INTEGER INDICATOR
Properties:
at r >0 > at r <0
7 . For any rational numbersr Ands from inequality > should
> at a >1 at
Abbreviated multiplication formulas.
Example 1. Simplify the expression.
Let's apply the properties of powers (multiplying powers with the same basis and division of powers with the same base): 
.
Answer: 9m 7 .
Example 2. Reduce a fraction:
Solution. So the domain of definition of the fraction is all numbers except x ≠ 1 and x ≠ -2. However, 
.By reducing the fraction, we get .The domain of definition of the resulting fraction: x ≠ -2, i.e. wider than the range of definition of the original fraction. Therefore, the fractions and are equal for x ≠ 1 and x ≠ -2.
Example 3. Reduce a fraction: 
Example 4. Simplify: 
Example 5.Simplify: 
Example 6. Simplify: 
Example 7. Simplify: 
Example 8. Simplify: 
Example 9. Calculate: 
.
Solution.
Example 10. Simplify the expression:
Solution. 
Example 11.Reduce a fraction if
Solution. 
.
Example 12. Free yourself from irrationality in the denominator of a fraction
Solution. In the denominator we have irrationality of the 2nd degree, therefore we multiply both the numerator and the denominator of the fraction by the conjugate expression, that is, the sum of the numbers and , then in the denominator we have the difference of squares, which eliminates the irrationality.
OPTION - I1. Simplify the expression:
, where a is a rational number,
b
– natural number
, 
5. Simplify:
; 
, 
, 
10. Follow this action:
8. Reduce the fraction 
9. Take action 
OPTION - II
1. Simplify the expression:
2. Find the meaning of the expression:
3. Imagine a degree with fractional indicator in the form of a root
4. Lead specified expression to mind , where a is a rational number,
b
– natural number
, 
5. Simplify:
; 
6. Replace arithmetic roots degrees with fractional exponent
, 
, 
7. Present the expression as a fraction whose denominator does not contain a root sign
10. Follow this action:
8. Reduce the fraction 
9. Take action 
1. Follow this action:
2. Find the meaning of the expression:
3. Represent a power with a fractional exponent as a root
4. Reduce the specified expression to the form , where a is a rational number,
b
– natural number
, 
5. Simplify:
; 
6. Replace arithmetic roots with powers with a fractional exponent
, 
, 
7. Present the expression as a fraction whose denominator does not contain a root sign
10. Follow this action:
8. Reduce the fraction 
9. Take action
OPTION - IV
1. Follow this action:
2. Find the meaning of the expression:
3. Represent a power with a fractional exponent as a root
, 
4. Reduce the specified expression to the form , where a is a rational number,
b
– natural number
, 
5. Simplify:
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