Rationalize the denominator of the following:

Rationalize the denominator of the following:

(i) 3/4√5
(ii) 5√7 / √3
(iii) 3/(4 – √7)
(iv) 17/(3√2 + 1)
(v) 16/ (√41 – 5)
(vi) 1/ (√7 – √6)
(vii) 1/ (√5 + √2)
(viii) (√2 + √3) / (√2 – √3)

Solution:

(i) 3/4√5
Let us rationalize,
3/4√5 = (3×√5) /(4√5×√5)
= (3√5) / (4×5)
= (3√5) / 20
(ii) 5√7 / √3
Let us rationalize,
5√7 / √3 = (5√7×√3) / (√3×√3)
= 5√21/3
(iii) 3/(4 – √7)
Let us rationalize,
3/(4 – √7) = [3×(4 + √7)] / [(4 – √7) × (4 + √7)]
= 3(4 + √7) / [4² – (√7)²]
= 3(4 + √7) / [16 – 7]
= 3(4 + √7) / 9
= (4 + √7) / 3
(iv) 17/(3√2 + 1)
Let us rationalize,
17/(3√2 + 1) = 17(3√2 – 1) / [(3√2 + 1) (3√2 – 1)]
= 17(3√2 – 1) / [(3√2)² – 1²]
= 17(3√2 – 1) / [9.2 – 1]
= 17(3√2 – 1) / [18 – 1]
= 17(3√2 – 1) / 17
= (3√2 – 1)
(v) 16/ (√41 – 5)
Let us rationalize,
16/ (√41 – 5) = 16(√41 + 5) / [(√41 – 5) (√41 + 5)]
= 16(√41 + 5) / [(√41)² – 5²]
= 16(√41 + 5) / [41 – 25]
= 16(√41 + 5) / [16]
= (√41 + 5)
(vi) 1/ (√7 – √6)
Let us rationalize,
1/ (√7 – √6) = 1(√7 + √6) / [(√7 – √6) (√7 + √6)]
= (√7 + √6) / [(√7)² – (√6)²]
= (√7 + √6) / [7 – 6]
= (√7 + √6) / 1
= (√7 + √6)
(vii) 1/ (√5 + √2)
Let us rationalize,
1/ (√5 + √2) = 1(√5 – √2) / [(√5 + √2) (√5 – √2)]
= (√5 – √2) / [(√5)² – (√2)²]
= (√5 – √2) / [5 – 2]
= (√5 – √2) / [3]
= (√5 – √2) /3
(viii) (√2 + √3) / (√2 – √3)
Let us rationalize,
(√2 + √3) / (√2 – √3) = [(√2 + √3) (√2 + √3)] / [(√2 – √3) (√2 + √3)]
= [(√2 + √3)²] / [(√2)² – (√3)²]
= [2 + 3 + 2√2√3] / [2 – 3]
= [5 + 2√6] / -1
= – (5 + 2√6)

More Solution:

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