Question

Find the rational values of a and b from each of the following:
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Answer 1

$1.\frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5}- \sqrt{3} } = a+b \sqrt{15}$

By Rationalization,

$L.H.S, \\\frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5}+ \sqrt{3} }$

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$\small \bold{On \: Nominator, \: by \: formula, \: (a+b)^2 =a^2+ b^2+2ab}$

$\small\bold{On \: Denominator, \: by \: formula, \: a^2 -b^2 =(a+b)(a-b)}$

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

$\frac{5 + 3 + 2 \sqrt{15} }{5-3}$

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

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Then,

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

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

$4 + \sqrt{15} = a+b \sqrt{15}$

Now,

a =4

b√15 =√15

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a= 4

b =1