Question

Please anyone answer quickly , it's really urgent.​

Answer 1

★$~~\large \sf {\underline {\pink{Given:}}}$

Given that : $\sf \dfrac{3}{4\sqrt{5}-\sqrt{3}}+\dfrac{2}{4\sqrt{5}+\sqrt{3}}=a\sqrt{5}+b\sqrt{3}$.

★$~~\large \sf {\underline{\pink{To~find:}}}$

We should find the value of a and b.

★$~~\large \sf {\underline{\pink{Solution:}}}$

Rationalising the denominator of $\sf \dfrac{3}{4\sqrt{5}-\sqrt{3}}$ :

$\implies \sf \dfrac{3}{4\sqrt{5}-\sqrt{3}}\times(\dfrac{4\sqrt{5}+\sqrt{3}}{4\sqrt{5}+\sqrt{3}})$

$\implies \sf \dfrac{3(4 \sqrt{5} + \sqrt{3} )}{(4 \sqrt{5} - \sqrt{3} )(4 \sqrt{5} + \sqrt{3} ) }$

$\implies \sf \dfrac{12 \sqrt{5} + 3 \sqrt{3} }{ {(4 \sqrt{5})^{2} - (\sqrt{3})^{2} } }$

$\implies \sf \dfrac{12 \sqrt{5} + 3 \sqrt{3} }{(16 \times 5) - (3)}$

$\orange{ \implies { \pmb{\sf \dfrac{12 \sqrt{5} + 3 \sqrt{3} }{77} \: \: ...(1) }}}$

Rationalising the denominator of $\sf \dfrac{2}{4\sqrt{5}+\sqrt{3}}$ :

$\implies \sf \dfrac{2}{4 \sqrt{5} + \sqrt{3} } \times ( \dfrac{4 \sqrt{5} - \sqrt{3} }{4 \sqrt{5} - \sqrt{3} } )$

$\implies \sf \dfrac{8 \sqrt{5} - 2 \sqrt{3} }{(4 \sqrt{5} + \sqrt{3})(4 \sqrt{5} - \sqrt{3}) }$

$\implies \sf \dfrac{8 \sqrt{5} - 2 \sqrt{3} }{ {(4 \sqrt{5}) }^{2} - {( \sqrt{3}) }^{2} }$

$\implies \sf \dfrac{8 \sqrt{5} - 2 \sqrt{3} }{(16 \times 5) - (3)}$

$\orange{ \implies \pmb{ \sf{ \dfrac{8 \sqrt{5} - 2 \sqrt{3} }{77} \: \: ...(2)}} }$

Now, adding (1) and (2) :

$\implies \sf (\dfrac{12 \sqrt{5} + 3 \sqrt{3} }{77}) + ( \dfrac{8 \sqrt{5} - 2 \sqrt{3} }{77} )$

$\implies \sf \dfrac{12 \sqrt{5} + 3 \sqrt{3} + 8 \sqrt{5} - 2 \sqrt{3} }{77}$

$\implies \sf \dfrac{12 \sqrt{5} + 8 \sqrt{5} + 3 \sqrt{3} - 2 \sqrt{3} }{77}$

$\orange{ \implies \pmb { \sf \dfrac{20 \sqrt{5} + \sqrt{3} }{77} }}$

———————————————

${\orange{\pmb{\sf{Therefore,~a=\dfrac{20}{77}~and~b=\dfrac{1}{77}~for ~a\sqrt{5}+b\sqrt{3}}.}}}$

Answer 2

Answer:

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