Question

If $a+\frac{1}{a}$ = 4, the value of $a^3+\frac{1}{a^3}$ is equal to .

• A. $56$
• B. $42$
• C. $52$
• D. $62$

Answer 1

The correct option is C $52$

Using the identity,
$(a+b{)}^3=a^3+b^3+3ab(a+b)$,
we have

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

$\Rightarrow$ $a^3+\frac{1}{a^3}$ = $(a+\frac{1}{a}{)}^3-3(a+\frac{1}{a})$

$\Rightarrow$ $a^3+\frac{1}{a^3}$ = $4^3-3\left(4\right)$
$\Rightarrow$ $a^3+\frac{1}{a^3}$ = $64-12$
$\Rightarrow$ $a^3+\frac{1}{a^3}$ = $52$