Question

If $\frac{a^2-1}{a}=2,$ then the value of $\frac{a^6-1}{a^3}$ is

Answer 1

Solve for the required value

On raising the given expression to the power of three on both sides,

We get,

$$\begin{gathered} \Rightarrow {\left(\frac{a^2-1}{a}\right)}^3={\left(2\right)}^3 \\ \Rightarrow \frac{a^6-1-3a^4+3a^2}{a^3}=8 \\ \Rightarrow \frac{a^6-1}{a^3}-\frac{3a^2\left(a^2-1\right)}{a^3}=8 \\ \Rightarrow \frac{a^6-1}{a^3}-6=8 \\ \Rightarrow \frac{a^6-1}{a^3}=8+6 \\ \Rightarrow \frac{a^6-1}{a^3}=14 \end{gathered}$$

Hence, the required answer is $\frac{a^6-1}{a^3}=14$