Question

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Q).What are the value of A and B respectively,if
$\frac{ \sqrt{5 - 1} }{ \sqrt{5 + 1} } + \frac{ \sqrt{5 + 1} }{ \sqrt{5 - 1} } = A+B \sqrt{5} ?$
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Solve with step by step.

Answer 1

Step-by-step explanation:

Question:

$\frac{ \sqrt{5 - 1} }{ \sqrt{5 + 1} } + \frac{ \sqrt{5 + 1} }{ \sqrt{5 - 1} } = A+B \sqrt{5} ?$

To find:

The value of A and B respectively.

Solution:

$\frac{\sqrt{{5}\mathrm{{-}}{1}}}{\sqrt{{5}\mathrm{{+}}{1}}}$=$\frac{\sqrt{{5}\mathrm{{-}}{1}}}{\sqrt{{5}\mathrm{{+}}{1}}}$×$\frac{\sqrt{{5}\mathrm{{-}}{1}}}{\sqrt{{5}\mathrm{{-}}{1}}}$

=$\frac{\mathrm{(}\sqrt{{5}\mathrm{{-}}{1}}{\mathrm{)}}^{2}}{\mathrm{(}\sqrt{{5}{\mathrm{)}}^{2}\mathrm{{-}}{\mathrm{(}}{1}}{\mathrm{)}}^{2}}$

=$\frac{{5}\mathrm{{+}}{1}\mathrm{{-}}{2}\sqrt{5}}{{5}\mathrm{{-}}{1}}$

=$\frac{{6}\mathrm{{-}}{2}\sqrt{5}}{4}$

=$\fbox{${\frac{{3}\mathrm{{-}}\sqrt{5}}{2}}$}$

$\frac{\sqrt{{5}\mathrm{{+}}{1}}}{\sqrt{{5}\mathrm{{-}}{1}}}$=$\frac{\sqrt{{5}\mathrm{{+}}{1}}}{\sqrt{{5}\mathrm{{-}}{1}}}$×$\frac{\sqrt{{5}\mathrm{{+}}{1}}}{\sqrt{{5}\mathrm{{+}}{1}}}$

=$\frac{\mathrm{(}\sqrt{{5}\mathrm{{+}}{1}{\mathrm{)}}^{2}}}{\mathrm{(}\sqrt{{5}{\mathrm{)}}^{2}\mathrm{{-}}{1}}}$

=$\frac{{5}\mathrm{{+}}{1}\mathrm{{+}}{2}\sqrt{5}}{{5}\mathrm{{-}}{1}}$

=$\frac{{6}\mathrm{{+}}{2}\sqrt{5}}{4}$

=$\fbox{${\frac{{3}\mathrm{{+}}\sqrt{5}}{2}}$}$

∴ $\frac{\sqrt{{5}\mathrm{{-}}{1}}}{\sqrt{{5}\mathrm{{+}}{1}}}$+$\frac{\sqrt{{5}\mathrm{{+}}{1}}}{\sqrt{{5}\mathrm{{-}}{1}}}$=A+B$\sqrt{5}$

=> $\frac{{3}\mathrm{{-}}\sqrt{5}}{2}$+$\frac{{3}\mathrm{{+}}\sqrt{5}}{2}$=A+B$\sqrt{5}$

=> $\frac{6}{2}$ = A+B$\sqrt{5}$

=> 3 = A+B$\sqrt{5}$

∴ A+B$\sqrt{5}$ = 3+(0)$\sqrt{5}$

=> A = 3 and B = 0

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