# Find the square root of the following complex numbers: 7 + 24i - Mathematics and Statistics

Sum

Find the square root of the following complex numbers: 7 + 24i

#### Solution

Let sqrt(7 + 24"i") = a + bi, where a, b ∈ R
Squaring on both sides, we get
7 + 24i = (a + bi)2
∴ 7 + 24i = a2 + b2i2 + 2abi
∴ 7 + 24i = (a2 – b2) + 2abi         ...[∵ i2 = – 1]
Equating real and imaginary parts, we get
a2 – b2 = 7 and 2ab = 24

∴ a2 – b2 = 7 and b = 12/"a"

∴ "a"^2 - (12/"a")^2 = 7

∴ "a"^2 - 144/"a"^2 = 7

∴ a4 – 144 = 7a2
∴ a4 – 7a2 – 144 = 0
∴ (a2 – 16)(a2 + 9) = 0
∴ a2 = 16 or a2 = – 9
But a ∈ R
∴ a2 ≠ – 9
∴ a2 = 16
∴ a = ± 4
When a = 4, b = 12/4 = 3

When a = – 4, b = 12/(-4) = – 3

∴ sqrt(7 + 24"i") = ± (4 + 3i).

Concept: Square Root of a Complex Number
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#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board
Chapter 3 Complex Numbers
Exercise 3.2 | Q 1. (ii) | Page 40