Q.22 Find the square root of \( 15-8 i \).

Question

Find the square root of \( 15-8 i \).

Answer 1

Equating the real and imaginary parts separately, we get,

Now x is a real number

\(\therefore\) the square roots of 15 - 8i are 4 - i and -4 + i, i.e., \(\pm (4-i)\).

Answer 2

Let \(\sqrt{15 - 8i}\) = x + iy, where x, y ∈ R.

On squaring both sides, we get,

15 – 8i = (x² – y²) + i2xy

x² – y² = 15 and 2xy = – 8

∴ y = \(- \frac 4x\)

∴ x² – \((- \frac 4x)^2\) = 15

∴ x² – \(\frac{16}{x^2}\) = 15

x⁴ – 16 = 15x²

x⁴ – 15x² – 16 = 0

(x² – 16)(x² + 1) = 0

∴ x² = 16 or x² = – 1

∴ x = ± 4

When x = 4, y = -1

When x = – 4, y = 1

The square roots of 15 – 8i are ± (4 – i).