Question

Choose the correct answer from the alternatives given :
Express $\frac{(4+\sqrt{15}{)}^{3/2}+(4-\sqrt{15}{)}^{3/2}}{(6+\sqrt{35}{)}^{3/2}+(6-\sqrt{35}{)}^{3/2}}$ in rational form.

• A. $\frac{7}{12}$
• B. $\frac{7}{26}$
• C. $\frac{7}{13}$
• D. None of these

Answer 1

The correct option is C $\frac{7}{13}$

$4+\sqrt{1}5=\frac{1}{2}(8+2\sqrt{1}5)$
$\frac{1}{2}(3+5+2\sqrt{3}\sqrt{5})=$ $\frac{1}{2}(\sqrt{3}+\sqrt{5}{)}^2$
$6+\sqrt{3}5=$$\frac{1}{2}(12+2\sqrt{35})$
$\frac{1}{2}(7+5+2\sqrt{7}\sqrt{5})=$$\frac{1}{2}(\sqrt{7}+\sqrt{5}{)}^2$ etc.
Hence given expression,
$\frac{{\left(\frac{1}{2}\right(\sqrt{5}+\sqrt{3}{)}^2)}^{3/2}+{\left(\frac{1}{2}\right(\sqrt{5}-\sqrt{3}{)}^2)}^{3/2}}{{\left(\frac{1}{2}\right(\sqrt{7}+\sqrt{5}{)}^2)}^{3/2}+{\left(\frac{1}{2}\right(\sqrt{7}-\sqrt{5}{)}^2)}^{3/2}}$
$\frac{(\sqrt{5}+\sqrt{3}{)}^3+(\sqrt{5}-\sqrt{3}{)}^3}{(\sqrt{7}+\sqrt{5}{)}^3-(\sqrt{7}-\sqrt{5}{)}^3}$
$\because (a+b{)}^3+(a-b{)}^3=2(a^3+3a{b}^2)$
$\because (a+b{)}^3-(a-b{)}^3=2(b^3+3a^{2}b)$
$\frac{2\left(\right(\sqrt{5}{)}^3+3\sqrt{5}\left(\sqrt{3}{)}^2\right)}{2\left(\right(\sqrt{5}{)}^3+3\sqrt{5}\left(\sqrt{7}{)}^2\right)}$
$=\frac{5\sqrt{5}+9\sqrt{5}}{5\sqrt{5}+21\sqrt{5}}=\frac{5+9}{5+21}=\frac{14}{26}=\frac{7}{13}$