Question

Rationalisation. Please help!

Answer 1

${x}^{2} = 11 + 2 \sqrt{30} \\ x = \sqrt{11 + 2 \sqrt{30} } \: \: \: \: \\ x = \sqrt{ {( \sqrt{5}) }^{2} + {( \sqrt{6}) }^{2} + 2( \sqrt{5})( \sqrt{6}) } \\ \\ = \sqrt{ {( \sqrt{5} + \sqrt{6} )}^{2} } \\ = \sqrt{5} + \sqrt{6}$


(i) Have already shown the answer for x above.

(ii)
$\frac{1}{x} = \frac{1}{ \sqrt{5} + \sqrt{6} }$
Rationalizing the term,
$\frac{1}{ \sqrt{5} + \sqrt{6} } \times \frac{ \sqrt{5} - \sqrt{6} }{ \sqrt{5} - \sqrt{6} } \\ = \frac{ \sqrt{5} - \sqrt{6} }{5 - 6} \\ = \frac{ \sqrt{5} - \sqrt{6} }{ (- 1)} \\ = \sqrt{6} - \sqrt{5}$

(iii) Adding (i) and (ii) will give:
$\sqrt{5} + \sqrt{6} + \sqrt{6} - \sqrt{5} \\ = 2 \sqrt{6}$
(iv) Subtraction (ii) from (i)

$\sqrt{5} + \sqrt{6} - ( \sqrt{6} - \sqrt{5} ) \\ = \sqrt{5} + \sqrt{6} - \sqrt{6} + \sqrt{5} \\ = 2 \sqrt{5}$



Tada!!

all done.