# PRACTICE PROBLEMS ON RATIONALIZING DENOMINATOR

Question 1 :

Rationalize the denominator

(i)  1/√50

Solution :

In order to rationalize the denominator, we have to multiply both numerator and denominator of the given rational number by √50.

(ii)  5/3√5

Solution :

=  5/3√5

Let us multiply the numerator and denominator by √5.

=  (5/3√5) ⋅ (√5/√5)

=  5√5/3(5)

=   √5/3

Hence the answer is √5/3.

(iii)  √75/√18

Solution :

=  √75/√18

= 5 √3/3√2

=  (5/3) (√3/2)

Multiply both numerator and denominator by √2, we get

=  5 √6/6

Hence the answer is √6/6.

(iv)  3√5/√6

Solution :

=  3√5/√6

= (3√5/√6) ⋅ (√6/√6)

=  3√30/6

=  √30/2

Hence the answer is √30/2.

Question 2 :

Rationalize the denominator and simplify

(i)  (√48 + √32) / (√27 - √18)

Solution :

Since the denominator is of 2 terms, we have to multiply the numerator and denominator by the conjugate of denominator.

=  [(√48 + √32) / (√27 - √18)] ⋅ [(√27+√18)/(√27+√18)]

=  (√48 + √32)(√27+√18) / (27 - 18)

=  (√(48  27) + √(48  18) + √(32  27) + √(32  18)/9

=  (36 + 12√6 + 12√6 + 24)/9

=  (60 + 24√6)/9

=  (20 + 8√6)/3

=  (4/3)(5 + 2√6)

(ii)  (5√3 + √2) / (√3 + √2)

Solution :

=  (5√3 + √2) / (√3 + √2)

=  [(5√3 + √2) / (√3 + √2)] ⋅ [(√3 - √2) / (√3 - √2)]

=  (5(3) - 5√6 + √6 - 2) / (3 - 2)

=  (15 - 6√6 - 2) / 1

=  13 - 6√6

(iii)  (2√6 - √5) / (3√5 - 2√6)

Solution :

=  (2√6 - √5) / (3√5 - 2√6)

(iv)  [√5/(√6 + 2)] - [√5/(√6 - 2)]

Solution :

=   [√5/(√6 + 2)] - [√5/(√6 - 2)]

=  [√5(√6 - 2) - √5(√6 + 2)]/(√6 - 2)(√6 + 2)

=  [√30 - 2√5 - √30 - 2√5)]/(6 - 4)

=  -4√5/2

=  -2√5

Question 3 :

Find the value of a and b if

Solution :

=  [(√7 - 2)/(√7 + 2)] ⋅ [(√7 - 2)/(√7 - 2)]

=  [(√7 - 2)2/(√7 + 2)(√7 - 2)]

=  (7 - 4√7 + 4)/(7 - 4)

=  (11 - 4√7)/3

=  (11/3) - (4√7/3)

Hence the value of a is 4/3 and b is 11/3

Question 4 :

If x = 5 + 2, then find the value of x2 + 1/x2

Solution :

x = √5 + 2

a2 + b2  =  (a + b)2 - 2ab

x+ (1/x)2  =  (x + (1/x))2 - 2x(1/x)

=  (x + (1/x))2 - 2  ----(1)

x + (1/x)  = √5 + 2 + (1/(√5 + 2))

=  ((√5 + 2)2 + 1)/(√5 + 2)

=  (5 + 2 + 4√5 + 1)/(2 + √5)

=  (8 + 4√5)/(2 + √5)

=  4(2 + √5)/(2 + √5)

=  4

x+ (1/x)2   =  4

By applying the value of x+ (1/x)2 in (1), we get

=  4 - 2

=  2

Hence 2 is the answer.

Question 5 :

Given that √2  =  1.414, find the value of (8-5√2)/(3-2√2) (to 3 places of decimals).

Solution :

(8-5√2)/(3-2√2)

=  5.414 is the answer

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