Question

(√5 -2 / √5 +5)/(√5 + 2/√5-2) = a + b√5 And a and b are rational numbers.

Answer 1

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

To find a and b.

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

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

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

$= \frac{5 + 3 \sqrt{5} }{20} .........(1)$

Now equating (1) and a+b√5 we get

$\frac{5 + 3 \sqrt{5} }{20} = a + b \sqrt{5}$

$= > a = \frac{5}{20} \: \: and \: \: b = \frac{ 3}{20}$

Or a= 1/4 and b= 3/20

Hope it clears your doubts

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