Question

5. Rationalise the denominators of the following: a) 1/(√5+√2)​

Answer 1

Step-by-step explanation:

$\frac{1}{( \sqrt{5} + \sqrt{2} ) } = \frac{1}{( \sqrt{5} + \sqrt{2} ) } \times \frac{( \sqrt{5} - \sqrt{2)} }{( \sqrt{5} - \sqrt{2)} }$

$\frac{1}{( \sqrt{5} + \sqrt{2} )} = \frac{ \sqrt{5} - \sqrt{2} }{( \sqrt{5} + \sqrt{2} )( \sqrt{5} - \sqrt{2}) }$

$\frac{1}{ \sqrt{5} + \sqrt{2} } = \frac{ \sqrt{5} - \sqrt{2} }{( \sqrt{5})^{2} - ( \sqrt{2)}^{2} }$

$\frac{1}{ \sqrt{5} + \sqrt{2} } = \frac{ \sqrt{5} - \sqrt{2} }{ {5}- {2} }$

$\frac{1}{ \sqrt{5} + \sqrt{2} } = \frac{ \sqrt{5} - \sqrt{2} }{3}$

Answer 2

Answer:

$\frac{\sqrt{5}-\sqrt{2} }{3 }$

Step-by-step explanation:

$\frac{1}{\sqrt{5}+\sqrt{2} }$

Multiplying both numerator and denominator by $\sqrt{5}-\sqrt{2}$,

$=\frac{1}{\sqrt{5}+\sqrt{2} }$ × $\frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}-\sqrt{2} }$

⇒ $\frac{(1)(\sqrt{5}-\sqrt{2}) }{(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2}) }$

⇒ $\frac{(1)(\sqrt{5}-\sqrt{2}) }{(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2}) }$

⇒ $\frac{\sqrt{5}-\sqrt{2} }{(\sqrt{5})^{2}-(\sqrt{2})^{2} }$  $($∵ $(a+b)(a-b)=a^{2}-b^{2})$

⇒ $\frac{\sqrt{5}-\sqrt{2} }{5-2 }$ $($∵ $(\sqrt{5})^{2}=5, (\sqrt{2})^{2}=2)$

⇒ $\frac{\sqrt{5}-\sqrt{2} }{3 }$