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Question: \[\sqrt[\mathbf{3}]{\mathbf{(61}\mathbf{-}\mathbf{46}\sqrt{\mathbf{5}}\mathbf{)}}\mathbf{=}\]...

(61−465)3=\sqrt[\mathbf{3}]{\mathbf{(61}\mathbf{-}\mathbf{46}\sqrt{\mathbf{5}}\mathbf{)}}\mathbf{=}

A

1−251 - 2\sqrt{5}

B

1−51 - \sqrt{5}

C

2−52 - \sqrt{5}

D

None of these

Answer

1−251 - 2\sqrt{5}

Explanation

Solution

61−4653=a−b\sqrt[3]{61 - 46\sqrt{5}} = a - \sqrt{b}

⇒ 61−465=(a−b)3=a3+3ab−(3a2+b)b61 - 46\sqrt{5} = (a - \sqrt{b})^{3} = a^{3} + 3ab - (3a^{2} + b)\sqrt{b}

⇒ 61=a3+3ab,465=(3a2+b)b61 = a^{3} + 3ab,46\sqrt{5} = (3a^{2} + b)\sqrt{b}⇒61=(a2+3b)a61 = (a^{2} + 3b)a,

2320=(3a2+b)b23\sqrt{20} = (3a^{2} + b)\sqrt{b}

Soa=1,b=20a = 1,b = 20. Therefore 61−4653=1−20=1−25\sqrt[3]{61 - 46\sqrt{5}} = 1 - \sqrt{20} = 1 - 2\sqrt{5}