Question

Find the fourth root of $124-32\sqrt{15}$.

• A. $\sqrt{5}+\sqrt{3}$
• B. $\sqrt{5}-\sqrt{3}$
• C. $\sqrt{31}-\sqrt{4}$
• D. $\sqrt{60}-\sqrt{64}$

Answer 1

The correct option is B $\sqrt{5}-\sqrt{3}$

$(124-32\sqrt{15})=8^2-2\times 8\left(2\sqrt{15}\right)+(2\sqrt{15}{)}^2$

$=(8-2\sqrt{15}{)}^2$

$={\left[\right(\sqrt{5}{)}^2-2\left(\sqrt{5}\right)\left(\sqrt{3}\right)+\left(\sqrt{3}{)}^2\right]}^2$

$=(\sqrt{5}-\sqrt{3}{)}^4$

$\therefore {[124-32\sqrt{15}]}^{1/4}=(\sqrt{5}-\sqrt{3})$ .

Therefore, option $B$ is correct.