Maharashtra State BoardHSC Commerce 11th
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Find the square root of: 2+23i - Mathematics and Statistics

Sum

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

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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
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