If (a+1 a )^2=3; then show that a^3+1 a^3 =0 - Vidyadip

Question

If $\left(a+\frac{1}{a}\right)^2=3$; then show that $a^3+\frac{1}{a^3}=0$

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Answer

$\left(a+\frac{1}{a}\right)^2=3 $
$\Rightarrow a+\frac{1}{a}=\sqrt{3}$
Now, $\left(a+\frac{1}{a}\right)^3$
$=a^3+\frac{1}{a^3}+3\left(a+\frac{1}{a}\right) $
$\Rightarrow(\sqrt{3})^3 $
$=a^3+\frac{1}{a^3}+3(\sqrt{3}) $
$\Rightarrow a^3+\frac{1}{a^3} $
$=3 \sqrt{3}-3 \sqrt{3} $
$=0$

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