Question

Find $a^3-\frac{1}{a^3}ifa-\frac{1}{a}=4$

Answer 1

$a-\frac{1}{a}=4\Rightarrow {(a-\frac{1}{a})}^3=(4{)}^3\Rightarrow a^3-\frac{1}{a^3}-3a\times \frac{1}{a}(a-\frac{1}{a})=64\Rightarrow a^3-\frac{1}{a^3}-3(4)=64[\because a-\frac{1}{a}=4]\Rightarrow a^3-\frac{1}{a^3}-12=64\Rightarrow a^3-\frac{1}{a^3}=64+12\Rightarrow a^3-\frac{1}{a^3}=76$.