If (x2 + 1/x2) = 51, Find the value of
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 $x^{3}-1 / x^{3}=364$

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