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Find the square root of the following complex number:

1 + 4√3 i

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Let \(\sqrt{1+4\sqrt{3}i}\) = a + bi, where a, b ∈ R.

Squaring on both sides, we get

1 + 4√3 i = (a + bi)2

1 + 4√3i = a2 + b2 i + 2abi

1 + 4√3i = (a2 – b2 ) + 2abi …..[∵ i2 = -1]

Equating real arid imaginary parts, we get

a2 – b2 = 1 and 2ab = 4√3

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