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Complex Numbers

√4 √5i - 1 eq...

Question

$\sqrt{4 \sqrt{5} i - 1}$ equals

\pm (- 2 + i \sqrt{2})

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\pm (- 2 - i \sqrt{5})

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\pm (2 - i \sqrt{5})

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\pm (2 + i \sqrt{5})

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Solution

The correct option is C $\pm (2 + i \sqrt{5})$
$\sqrt{4 \sqrt{5} i - 1} = a + i b$
Squaring on both sides and comparing real and imaginary parts gives
$\Rightarrow a^{2} - b^{2} = - 1$ and $a b = 2 \sqrt{5}$
$\Rightarrow a = 2$ and $b = \sqrt{5}$ or $a = - 2$ and $b = - \sqrt{5}$
$∴ \sqrt{4 \sqrt{5} i - 1} = \pm (2 + \sqrt{5} i)$
Hence, option D.

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