# Find the square root of: 2+23i - Mathematics and Statistics

Sum

Find the square root of: 2 + 2 sqrt(3)"i"

#### Solution

Let sqrt(2 + 2sqrt(3)"i") = a + bi, where a, b ∈ R.
Squaring on both sides, we get
2 + 2 sqrt(3)"i" = a2 + b2i2 + 2abi
∴ 2 + 2 sqrt(3)"i" = a2 – b2 + 2abi      ...[∵ i2 = – 1]
Equating real and imaginary parts, we get
a2 – b2 = 2 and 2ab = 2sqrt(3)

∴ a2 – b2 = 2 and b = sqrt(3)/"a"

∴ "a"^2 - (sqrt(3)/"a")^2 = 2

∴ "a"^2 - 3/"a"^2 = 2

∴ a4 – 3 = 2a2
∴ a4 – 2a2 – 3 = 0
∴ (a2 – 3)(a2 + 1) = 0
∴ a2 = 3 or a2 = – 1
But a ∈ R
∴ a2 ≠ – 1
∴ a2  = 3
∴ a = ± sqrt(3)

When a = sqrt(3), "b" = sqrt(3)/sqrt(3) = 1

When a = -sqrt(3), "b" = sqrt(3)/-sqrt(3) = – 1

∴ sqrt(2 + 2sqrt(3)"i") = ± (sqrt(3) + "i").

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
Miscellaneous Exercise 3 | Q 6. (iii) | Page 43