Question

find 4+3√(5)÷4-3√(5)=a+b√(5)​

Answer 1

$\frac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} } = a+ b\sqrt{5} \\ \\ Solving \: LHS, \\ \\{ \large{ \boxed{ \frac{ {(4 + 3 \sqrt{5} )}^{2} }{(4 - 3 \sqrt{5} )(4 + 3 \sqrt{5} )}}}} \\ \\ {\large{ \boxed{ \frac{16 + 45 + 24 \sqrt{5} }{16 - 45} }}} \\ \\ {\large{ \boxed{ \frac{61 + 24 \sqrt{5} }{ - 29} }}}$

Hence,

${\huge{ \boxed{a = \frac{ - 61}{29} }}} \\ \\ {\huge{ \boxed{b = \frac{24}{29} }}}$

Answer 2

Question

${ \sf{ \dfrac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5}} = a + b \sqrt{5}}}$

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Solution

$\longrightarrow{ \sf{ \dfrac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5}} = a + b \sqrt{5}}}$

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

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$\longrightarrow{ \sf{ \dfrac{16 + 45 + 25 \sqrt{5} }{16 - 45}}}$

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$\longrightarrow{ \sf{ \dfrac{-61 + -24 \sqrt{5} }{29}}}$

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⠀$\longrightarrow{ \sf{ \dfrac{-61}{29}}-{\dfrac{-24 \sqrt{5}}{29}}}$⠀⠀⠀⠀⠀⠀⠀⠀

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Therefore,

$\longrightarrow{\sf{a = \dfrac{-61}{29}} } \\ \\ \\ \longrightarrow{\sf{b = \frac{-24}{29}}}$