Question

Express with rational denominator i) 1÷√6+√7+√13

Answer 1

1/(√7+√6-√13)

Rationalising factor = √7+√6+√13

→ 1/(√7+√6-√13) × √7+√6+√13/(√7+√6+√13)

→ √7+√6+√13/(√7+√6-√13)(√7+√6+√13)

→ √7+√6+√13/(√7+√6)²-(√13)²

→ √7+√6+√13/(√7²+√6²+2(√7)(√6))-13

→ √7+√6+√13/13+2√42-13

→ √7+√6+√13/2√42

The denominator is still irrational. So we have to rationalise it further

Now rationalising factor = √42

→ √7+√6+√13/2√42 × √42/√42

→ √42(√7+√6+√13)/2(√42)²

→ √42×7+√42×6+√42×13/2(42)

→ (7√6+6√7+√546)/84

→ 7√6/84 + 6√7/84 + √546/84

→ √6/12 + √7/14 + √546/84

Now the denominator is rationalised.

____________________
Hope It Helps You

Answer 2

Hello,

Solution:

$\frac{1}{ \sqrt{6} + \sqrt{7} + \sqrt{13} } \\ \\ = \frac{1}{ \sqrt{6} + \sqrt{7} + \sqrt{13} } \times \frac{ \sqrt{6} + \sqrt{7} - \sqrt{13} }{\sqrt{6} + \sqrt{7} - \sqrt{13}} \\ \\ = \frac{\sqrt{6} + \sqrt{7} - \sqrt{13}}{( { \sqrt{6} + \sqrt{7}) }^{2} - ( { \sqrt{13} })^{2} } \\ \\ = \frac{\sqrt{6} + \sqrt{7} - \sqrt{13}}{6 + 7 + 2 \sqrt{42} - 13 } \\ \\ = \frac{\sqrt{6} + \sqrt{7} - \sqrt{13}}{2 \sqrt{42} } \times \frac{ \sqrt{42} }{ \sqrt{42} } \\ \\ = \frac{6 \sqrt{7} + 7 \sqrt{6} - \sqrt{13 \times 42} }{2 \times 42}$
is the final answer