Question

Answer the following:

Find the square root of 18i

Answer 1

Let `sqrt(18"i")` = x + yi, where x, y ∈ R

On squaring both sides, we get,

18i = (x + yi)² = x² + y²i² + 2xyi

∴ 0 + 18i = (x² – y²) + 2xyi    ...[∵ i² = – 1]

Equating the real and imaginary parts separately, we get,

x² – y² = 0 and 2xy = 18

∴ y = `9/x`

∴ `x^2 - (9/x)^2` = 0

∴ `x^2 - 81/x^2` = 0

∴ x⁴ – 81 = 0

∴ (x² – 9)(x² + 9) = 0

∴ x² = 9 or x² = – 9

Now, x is a real number

∴ x² ≠ – 9

∴ x² = 9

∴ x = ± 3

When x = 3, y = `9/3` = 3

When x = – 3, y = `9/(-3)` = – 3

∴ the square roots of 18i are

3 + 3i and – 3 – 3i, i.e., ± (3 + 3i).