Question

If $x+\frac{1}{x}=5$ find $x^3+\frac{1}{x^3}$

Answer 1

Formula to be used : $(a+b{)}^3=a^3+b^3+3ab(a+b)$
$x+\frac{1}{x}=5$
Cubing on both sides:

$\to (x+\frac{1}{x}{)}^3=5^3$

$\to x^3+\frac{1}{x^3}+3\cdot x\cdot \frac{1}{x}\cdot (x+\frac{1}{x})=125$
$\to x^3+\frac{1}{x^3}+15=125$

$\to x^3+\frac{1}{x^3}=125-15=110$