# HOW TO FIND SQUARE ROOT OF RATIONAL EXPRESSION

How to Find Square Root of Rational Expression :

Here we are going to see some example problems on finding square root of rational expressions.

## How to Find Square Root of Rational Expression - Questions

Question 1 :

Find the square root of the following rational expressions.

(i) 400 xy12 z16 / 100 x8 y4 z4

Solution :

=  √(400 xy12 z16/ 100 x8 y4 z4)

=  √(4 yz12/x4 )

=  |(2 yz6)/x2|

(ii)  (7x2 + 2 √(14)x + 2)/(x2 - x/2 + 1/16)

Solution :

=  √[(7x2 + 2 √(14)x + 2)/(x2 - x/2 + 1/16)]

(7x2 + 2 √(14)x + 2)

First let us find the factors of the quadratic equation in the numerator. (x2 - x/2 + 1/16)

=  (4x + 1) (4x + 1)

=  √[(7x + √14)(7x + √14) / (4x + 1) (4x + 1)]

=  |(7x + √14)/(4x + 1)|

(iii)  121 (a + b)8 (x + y)8 (b - c)8

Solution :

=  √[121 (a + b)8 (x + y)8 (b - c)8]

=  √[11 ⋅ 11(a + b)4 (a + b)4 (x +y)4(x + y)4 (b - c)4 (b - c)4]

=  11|(a + b)4 (x + y)4 (b - c)4|

## Finding Square Root of Polynomial

Question 1 :

Find the square root of the following

(i) 4x2 + 20x + 25

Solution :

=  √(4x2 + 20x + 25)

Since it is in the form of quadratic equation, let us try to represent in the form of a2 + 2ab + b2

=  √((2x)2 + 2(2x)(5) + 52)

=  (2x + 5)2

=  |2x + 5|

Hence the square root of given quadratic polynomial is (2x + 5).

Alternative method :

=  √(4x2 + 20x + 25)

=  √(4x2 + 10x + 10x + 25)

=  √2x(2x + 5) + 5(2x + 5)

=  √(2x + 5) (2x + 5)

=  |2x + 5|

Let us look into the next example on "How to Find Square Root of Rational Expression".

(ii) 9x2 −24xy + 30xz −40yz +25z2 +16y2

Solution :

=    √(9x2 −24xy + 30xz −40yz +25z2 +16y2)

=    √(9x2 + 25z2 + 16y−24xy + 30xz −40yz)

=    √(3x)2 + (5z)2 + (-4y)2−2(3x)(-4y)+2(3x)(5z)−2(-4y)(5z))

=    √(3x)2 + (5z)2 + (-4y)2−2(3x)(2y)+2(3x)(5z)−2(-4y)(5z))

=  √(3x - 4y + 5z)2

=  |(3x - 4y + 5z)|

(iii)  1 + 1/x6 + 2/x3

Solution :

√(1 + 1/x6 + 2/x3)

=   √(1 + 2⋅1⋅(1/x3) + (1/x3)2)

=   √(1 + (1/x3))2

=  |1 + (1/x3)|

Hence the square root of given polynomial is 1 + (1/x3).

(iv) (4x2 − 9x + 2) (7x2 - 13x - 2) (28x2 - 3x - 1).

Solution :

=  (4x2 − 9x + 2) (7x2 - 13x - 2) (28x2 - 3x - 1)

4x2 − 9x + 2  =  (4x - 1)(x - 2)

7x2 - 13x - 2  =  (7x + 1)(x - 2)

28x2 - 3x - 1  =  (4x - 1) (7x + 1)

=  (4x - 1)(x - 2)(7x + 1)(x - 2)(4x - 1) (7x + 1)

=  |(4x - 1)(x - 2)(7x + 1)|

(v)  (2x2 + (17x/6) + 1)((3/2)x2 + 4x + 2)((4/3)x2 + (11x/3) + 2)

Solution :

=  (2x2 + (17x/6) + 1)((3/2)x2 + 4x + 2)((4/3)x2 + (11x/3) + 2)

(2x2 + (17x/6) + 1)  =  (12x2 + 17x + 6)/6

=  (3x + 2)(4x + 3)/6

((3/2)x2 + 4x + 2)  =  (3x2 + 8x + 4)/2

=  (x + 2)(3x + 2)/2

(4/3)x2 + (11x/3) + 2)  =  (4x2 + 11x + 6)/3

=  (x + 2)(4x + 3)/3

=  (3x + 2)(4x + 3)/6 ⋅ (x + 2)(3x + 2)/2 ⋅ (x + 2)(4x + 3)/3

=  |(3x + 2)(4x + 3)(x + 2)|/6 After having gone through the stuff given above, we hope that the students would have understood, "How to Find Square Root of Rational Expression".

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