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Question: \[\frac{\lbrack 4 + \sqrt{(15)}\rbrack^{3/2} + \lbrack 4 - \sqrt{(15)}\rbrack^{3/2}}{\lbrack 6 + \sq...

[4+(15)]3/2+[4−(15)]3/2[6+(35)]3/2−[6−(35)]3/2=\frac{\lbrack 4 + \sqrt{(15)}\rbrack^{3/2} + \lbrack 4 - \sqrt{(15)}\rbrack^{3/2}}{\lbrack 6 + \sqrt{(35)}\rbrack^{3/2} - \lbrack 6 - \sqrt{(35)}\rbrack^{3/2}} =

A

1

B

7/13

C

13/7

D

None of these

Answer

7/13

Explanation

Solution

Let 4+15=x4 + \sqrt{15} = x, then 4−15=1x4 - \sqrt{15} = \frac{1}{x}

6+35=y6 + \sqrt{35} = y, then 6−35=1y6 - \sqrt{35} = \frac{1}{y}

∴\therefore Given expression = x3/2+1x3/2y3/2−1y3/2=x3+1y3−1.(yx)3/2\frac{x^{3/2} + \frac{1}{x^{3/2}}}{y^{3/2} - \frac{1}{y^{3/2}}} = \frac{x^{3} + 1}{y^{3} - 1}.\left( \frac{y}{x} \right)^{3/2}

=(4+15)3+1(6+35)3−1.(6+354+15)3/2= \frac{(4 + \sqrt{15})^{3} + 1}{(6 + \sqrt{35})^{3} - 1}.\left( \frac{6 + \sqrt{35}}{4 + \sqrt{15}} \right)^{3/2}

=(4+15+1){(4+15)2−(4+15)+1}(6+35−1){(6+35)2+(6+35)+1}×(6+354+15)3/2= \frac{(4 + \sqrt{15} + 1)\{(4 + \sqrt{15})^{2} - (4 + \sqrt{15}) + 1\}}{(6 + \sqrt{35} - 1)\{(6 + \sqrt{35})^{2} + (6 + \sqrt{35}) + 1\}} \times \left( \frac{6 + \sqrt{35}}{4 + \sqrt{15}} \right)^{3/2}

=5+155+35⥂.{31+815−4−15+1}{71+1235+6+35+1}.(6+354+15)3/2= \frac{5 + \sqrt{15}}{5 + \sqrt{35}} ⥂ .\frac{\{ 31 + 8\sqrt{15} - 4 - \sqrt{15} + 1\}}{\{ 71 + 12\sqrt{35} + 6 + \sqrt{35} + 1\}}.\left( \frac{6 + \sqrt{35}}{4 + \sqrt{15}} \right)^{3/2}

=5+35+7×28+71578+1335(6+354+15)3/2= \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{7}} \times \frac{28 + 7\sqrt{15}}{78 + 13\sqrt{35}}\left( \frac{6 + \sqrt{35}}{4 + \sqrt{15}} \right)^{3/2}

=5+35+7.713.6+354+15= \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{7}}.\frac{7}{13}.\sqrt{\frac{6 + \sqrt{35}}{4 + \sqrt{15}}}

=713.3+55+7.(5+7)22.2(3+5)2= \frac{7}{13}.\frac{\sqrt{3} + \sqrt{5}}{\sqrt{5} + \sqrt{7}}.\sqrt{\frac{(\sqrt{5} + \sqrt{7})^{2}}{2}.\frac{2}{(\sqrt{3} + \sqrt{5})^{2}}}

=713.3+55+7.5+73+5=713= \frac{7}{13}.\frac{\sqrt{3} + \sqrt{5}}{\sqrt{5} + \sqrt{7}}.\frac{\sqrt{5} + \sqrt{7}}{\sqrt{3} + \sqrt{5}} = \frac{7}{13}.