Question:
If $\left(x^{2}+1 / x^{2}\right)=51$, Find the value of $x^{3}-1 / x^{3}$
Solution:
Given, $\left(x^{2}+1 / x^{2}\right)=51$
We know that, $(x-y)^{2}=x^{2}+y^{2}-2 x y \ldots 1$
Substitute $\left(x^{2}+1 / x^{2}\right)=51$ in eq 1
$(x-1 / x)^{2}=x^{2}+1 / x^{2}-2^{*} x^{*} 1 / x$
$(x-1 / x)^{2}=x^{2}+1 / x^{2}-2$
$(x-1 / x)^{2}=51-2$
$(x-1 / x)^{2}=49$
$(x-1 / x)=\sqrt{49}$
$(x-1 / x)=\pm 7$
We need to find $x^{3}-1 / x^{3}$
So, $a^{3}-b^{3}=(a-b)\left(a^{2}+b^{2}+a b\right)$
$x^{3}-1 / x^{3}=(x-1 / x)\left(x^{2}+1 / x^{2}+(x * 1 / x)\right.$
We know that,
$(x-1 / x)=7$ and $\left(x^{2}+1 / x^{2}\right)=51$
$x^{3}-1 / x^{3}=7(51+1)$
$x^{3}-1 / x^{3}=7(52)$
$x^{3}-1 / x^{3}=364$
Hence, the value of $x^{3}-1 / x^{3}=364$