Question

If $(4+\sqrt{15}{)}^{\frac{3}{2}}-(4-\sqrt{15}{)}^{\frac{3}{2}}=k\sqrt{6},$ then $k=$

• A. 9
• B. 8
• C. 7
• D. 6

Answer 1

The correct option is A 9

(A) Given,
$(4+\sqrt{15}{)}^{3/2}-(4-\sqrt{15}{)}^{3/2}=k\sqrt{6}$
$\Rightarrow k\sqrt{6}=(\sqrt{4+\sqrt{15}}{)}^3-(4-\sqrt{15}{)}^3$
$=\frac{1}{2\sqrt{2}}\left[\right(\sqrt{8+2\sqrt{15}}{)}^3-\left(\sqrt{8-2\sqrt{15}}{)}^3\right]$
$=\frac{1}{2\sqrt{2}}\left[\right(\sqrt{5}+\sqrt{3}{)}^3-(\sqrt{5}-\sqrt{3}{)}^3]$
$=\frac{1}{2\sqrt{2}}\times 36\sqrt{3}=\frac{1}{4}\times 36\sqrt{3}=9\sqrt{6}$

$\therefore k=9$