PRACTICE QUESITONS ON RATIONALIZING THE DENOMINATOR

Question 1 :

Rationalize the denominator

1/√50

Solution :

= 1/√50

Since we have only one term at the denominator, by multiplying both numerator and denominator by √50, we may rationalize the denominator.

= (1/√50)(√50/√50)

= √50/50

= √(5 x 5 x 2) / 50

= 5 √2/50

= √2/10

Question 2 :

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.

Question 3 :

√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 5 √6/6.

Question 4 :

3√5/√6

Solution :

  =  3√5/√6

  = (3√5/√6) ⋅ (√6/√6)

=  3√30/6

=  √30/2

Hence the answer is √30/2.

Question 5 :

Rationalize the denominator and simplify

(√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)

Question 6 :

 (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

Question 7 :

(2√6 - √5) / (3√5 - 2√6)

Solution :

=  (2√6 - √5) / (3√5 - 2√6)

Conjugate of the denominator 3√5 - 2√6 is 3√5 + 2√6.

Multiplying both numerator and denominator by the conjugate, we get

= [(2√6 - √5) / (3√5 - 2√6)] [(3√5 + 2√6)/ (3√5 + 2√6)]

= (2√6 - √5)(3√5 + 2√6) / (3√5 - 2√6)(3√5 + 2√6)

Using distributive property for the numerator and simplifying, we get

= 2√6(3√5) + (2√6) (2√6) - √5(3√5) - √5(2√6)

= 6√30 + 4(6) - 3(5) - 2√30

= 4√30 + 24 - 15

= 4√30 + 9

Using algebraic identity, simplifying the 

(3√5 - 2√6)(3√5 + 2√6)

= (3√5)² - (2√6)²

= 9(5) - 4(6)

= 45 - 24

= 21

= (4√30 + 9)/21

Question 8 :

 [√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 9 :

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 10 :

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

Solution :

x = √5 + 2

a² + b²  =  (a + b)² - 2ab 

x² + (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

x² + (1/x)²   =  4

By applying the value of x² + (1/x)² in (1), we get

  =  4 - 2 

  =  2

Hence 2 is the answer.

Question 11 :

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)

Conjugate of the denominator = 3 + 2√2

= [(8 - 5√2)/(3 - 2√2)] [(3 + 2√2)/(3 + 2√2)]

= (8 - 5√2)(3 + 2√2) / [(3 - 2√2) (3 + 2√2)]

= [24 + 16√2 - 15√2 - 10√2√2] / [3² - (2√2)²]

= [24 + √2 - 10(2)] / [9 - (2²(2))]

= [24 + √2 - 20] / [9 - 8]

= 4 + √2

= 5 + 1.414

= 5.414

Question 12 :

The distance 𝐷, in kilometers, from the point of sight to the horizon is given by the formula 𝐷 = 4√H, where 𝑀 denotes the height of the point of sight above the sea level, in meters. To the nearest tenth of a kilometer, how far away is the horizon for a 180 cm tall man standing on a 40-m high cliff?

Solution :

𝐷 = 4√H

height o the man in meters,  

180 cm = 1.8 m

40 + 1.8

= 41.8

𝐷 = 4√41.8

= 4(6.46)

= 25.84

Question 13 :

When solving one of the trigonometry problems, a student come up with the answer (√3 − 1)/(1 + √3) answer to this problem was 2 − √3 . The textbook . Was the student’s answer equivalent to the textbook answer?

Solution :

= (√3 − 1)/(1 + √3)

Conjugate of the denominator = 1 - √3

= (√3 − 1)/(1 + √3) [(1 - √3)/(1 - √3)]

= -(1 - √3)(1 - √3)/(1 + √3)(1 - √3)

= -(1 - √3)²/(1 - √32)

= 1 - 2√3 + 

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