#### 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) / [4^{2} – (√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} – 1^{2}]

= 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} – 5^{2}]

= 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)

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