Question

Rationalise the denominator
$\frac{1}{ \sqrt{10} + \sqrt{14} + \sqrt{15} + \sqrt{21} }$
​

Answer 1

Answer:

$\frac{ \sqrt{10} + \sqrt{21} - \sqrt{14} - \sqrt{15} }{2}$

Step-by-step explanation:

Making suitable groups, we get

$\frac{1}{( \sqrt{10} + \sqrt{21} ) + ( \sqrt{14} + \sqrt{15} ) } \\ = \frac{1}{( \sqrt{10} + \sqrt{21} ) + ( \sqrt{14} + \sqrt{15} ) } \times \frac{( \sqrt{10} + \sqrt{21} ) - ( \sqrt{14} + \sqrt{15} ) }{( \sqrt{10} + \sqrt{21}) - ( \sqrt{14} + \sqrt{15} )} \\ = \frac{ \sqrt{10} + \sqrt{21} - \sqrt{14} - \sqrt{15} }{( \sqrt{10} + \sqrt{21})^{2} - ( \sqrt{14} + \sqrt{15})^{2} } \\ = \frac{ \sqrt{10} + \sqrt{21} - \sqrt{14} - \sqrt{15} }{10 + 21 + 2 \sqrt{210} - 14 - 15 - 2 \sqrt{210} } \\ = \frac{ \sqrt{10} + \sqrt{21} - \sqrt{14} - \sqrt{15} }{31 - 29} \\ = \frac{ \sqrt{10} + \sqrt{21} - \sqrt{14} - \sqrt{15} }{2}$

Answer 2

Answer:

Making suitable groups, we get

$\frac{1}{( \sqrt{10} + \sqrt{21} ) + ( \sqrt{14} + \sqrt{15} ) }$

$= \frac{1}{( \sqrt{10} + \sqrt{21} ) + ( \sqrt{14} + \sqrt{15} ) } \times \frac{( \sqrt{10} + \sqrt{21} ) - ( \sqrt{14} + \sqrt{15} ) }{( \sqrt{10} + \sqrt{21}) - ( \sqrt{14} + \sqrt{15} )}$ $= \frac{ \sqrt{10} + \sqrt{21} - \sqrt{14} - \sqrt{15} }{( \sqrt{10} + \sqrt{21})^{2} - ( \sqrt{14} + \sqrt{15})^{2} }$ $= \frac{ \sqrt{10} + \sqrt{21} - \sqrt{14} - \sqrt{15} }{10 + 21 + 2 \sqrt{210} - 14 - 15 - 2 \sqrt{210} }$

$= \frac{ \sqrt{10} + \sqrt{21} - \sqrt{14} - \sqrt{15} }{31 - 29}$

$= \frac{ \sqrt{10} + \sqrt{21} - \sqrt{14} - \sqrt{15} }{2}$