Question

If $a^2+\frac{1}{a^2}=20$ and $a^3+\frac{1}{a^3}=30$, find $a+\frac{1}{a}$ =

Answer 1

Given : $a^2+\frac{1}{a^2}=20$ and $a^3+\frac{1}{a^3}=30$

Now, $a^3+\frac{1}{a^3}=a^3+{\left(\frac{1}{a}\right)}^3$

$=(a+\frac{1}{a})(a^2+\frac{1}{a^2}-a\times \frac{1}{a})$

$=(a+\frac{1}{a})(a^2+\frac{1}{a^2}-1)$

$=(a+\frac{1}{a})(20-1)=(a+\frac{1}{a})19$

$⟹a^3+\frac{1}{a^3}=(a+\frac{1}{a})19$

$⟹30=(a+\frac{1}{a})19$

$⟹(a+\frac{1}{a})=\frac{30}{19}$