Question

If
$\frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } = x + y \sqrt{15}$
find the value of x and y​

Answer 1

$\frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } = x + y \sqrt{15}$

Consider Left Hand Side

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

The simplest Rationalising factor of √5 - √3 is √5 + √3. So, multiply the numerator and denominator of the given fraction by Rationalising factor.

$= \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} }$

$= \frac{ {( \sqrt{5} + \sqrt{3}) }^{2} }{ {( \sqrt{5}) }^{2} - {( \sqrt{3}) }^{2} }$

Since (x + y)(x - y) = x² - y²

$= \frac{ {( \sqrt{5})}^{2} + 2( \sqrt{5})( \sqrt{3}) + {( \sqrt{3})}^{2} }{5 - 3}$

Since (x + y)² = x² + 2xy + y²

$= \frac{5 + 2( \sqrt{5 \times 3}) + 3 }{2}$

$= \frac{8 + 2 \sqrt{15} }{2}$

$= \frac{2(4 + \sqrt{15}) }{2}$

$= 4 + \sqrt{15}$

$now \: consider \\ \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } =x + y \sqrt{15}$

$i.e \: 4 + \sqrt{15} = x + y \sqrt{15}$

Equating corresponding rational and irrational factors, we have

x = 4

y√15 = √15

=> y = 1

$\boxed{\tt{x = 4}}$

$\boxed{\tt{y = 1}}$