Simplify each of the following by rationalizing the denominator:

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Simplify each of the following by rationalizing the denominator:

(i) (7 + 3√5) / (7 – 3√5)
(ii) (3 – 2√2) / (3 + 2√2)
(iii) (5 – 3√14) / (7 + 2√14)

Solution:

(i) (7 + 3√5) / (7 – 3√5)
Let us rationalize the denominator, we get
(7 + 3√5) / (7 – 3√5) = [(7 + 3√5) (7 + 3√5)] / [(7 – 3√5) (7 + 3√5)]
= [(7 + 3√5)2] / [72 – (3√5)2]
= [72 + (3√5)2 + 2.7. 3√5] / [49 – 9.5]
= [49 + 9.5 + 42√5] / [49 – 45]
= [49 + 45 + 42√5] / [4]
= [94 + 42√5] / 4
= 2[47 + 21√5]/4
= [47 + 21√5]/2
(ii) (3 – 2√2) / (3 + 2√2)
Let us rationalize the denominator, we get
(3 – 2√2) / (3 + 2√2) = [(3 – 2√2) (3 – 2√2)] / [(3 + 2√2) (3 – 2√2)]
= [(3 – 2√2)2] / [32 – (2√2)2]
= [32 + (2√2)2 – 2.3.2√2] / [9 – 4.2]
= [9 + 4.2 – 12√2] / [9 – 8]
= [9 + 8 – 12√2] / 1
= 17 – 12√2
(iii) (5 – 3√14) / (7 + 2√14)
Let us rationalize the denominator, we get
(5 – 3√14) / (7 + 2√14) = [(5 – 3√14) (7 – 2√14)] / [(7 + 2√14) (7 – 2√14)]
= [5(7 – 2√14) – 3√14 (7 – 2√14)] / [72 – (2√14)2]
= [35 – 10√14 – 21√14 + 6.14] / [49 – 4.14]
= [35 – 31√14 + 84] / [49 – 56]
= [119 – 31√14] / [-7]
= -[119 – 31√14] / 7
= [31√14 – 119] / 7

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