[3] (61 - 46 5 ) = - Vidyadip

MCQ

$\sqrt[3]{{(61 - 46\sqrt 5 )}} = $

✓
$1 - 2\sqrt 5 $
B
$1 - \sqrt 5 $
C
$2 - \sqrt 5 $
D
None of these

✓

Answer

Correct option: A.

$1 - 2\sqrt 5 $

a
(a) $\sqrt[3]{{61 - 46\sqrt 5 }} = a - \sqrt b $

==> $61 - 46\sqrt 5 = {(a - \sqrt b )^3} = {a^3} + 3ab - (3{a^2} + b)\sqrt b $

==> $61 = {a^3} + 3ab,\,46\sqrt 5 = (3{a^2} + b)\,\sqrt b $

==> $61 = ({a^2} + 3b)\,a$, $23\sqrt {20} = (3{a^2} + b)\sqrt b $

So,$a = 1,\,\,b = 20$.

Therefore, $\sqrt[3]{{61 - 46\sqrt 5 }} = 1 - \sqrt {20} = 1 - 2\sqrt 5 $.

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