Question

If $x-\frac{1}{x}=\sqrt{3}$, then find the value of $x^3-\frac{1}{x^3}$.

• A. $7\sqrt{3}$
• B. $6\sqrt{3}$
• C. $3$
• D. $\sqrt{3}$

Answer 1

The correct option is B

$6\sqrt{3}$

$x-\frac{1}{x}=\sqrt{3}$ ......(i)
$x^3-\frac{1}{x^3}=(x-\frac{1}{x})(x^2+\frac{1}{x^2}+x\times \frac{1}{x})=(x-\frac{1}{x})(x^2+\frac{1}{x^2}+1)$..........(ii)
Squaring both sides of (i)
$(x-\frac{1}{x}{)}^2=(\sqrt{3}{)}^2{x}^2+\frac{1}{x^2}-2.x.\frac{1}{x}=3\Rightarrow x^2+\frac{1}{x^2}-2=3\Rightarrow x^2+\frac{1}{x^2}=5$
Substitute in (ii) to get
$x^3-\frac{1}{x^3}=\sqrt{3}(5+1)=6\sqrt{3}$