Question

if p+1/p=4, prove that p3+1/p3=52​

Answer 1

Suitable identity is $(a+b)^3=a^3+3a^2b+3ab^2+b^3$

Here $a=p$, $b=\frac{1}{p}$

$(p+\frac{1}{p} )^3=p^3+3p+\frac{3}{p} +\frac{1}{p^3}$

$(p+\frac{1}{p} )^3=p^3+\frac{1}{p^3}+3(p+\frac{1}{p} )$

Put the value.

$4^3=p^3+\frac{1}{p^3}+3(4)$

$p^3+\frac{1}{p^3}=4^3-3(4)$

∴ $52$ is $p^3+\frac{1}{p^3}$.

Hence shown.