√-7+24i

1.	√-7+24i
Answer» Let, (a + ib)²= - 7 + 24i Now using, (a + b)² = a² + b² + 2ab a² + (bi)² + 2abi = -7 + 24i Since i² = -1 a² - b² + 2abi = -7 + 24i Now, separating real and complex parts, we get ⇒ a² - b² = - 7…………..eq.1 ⇒ 2ab = 24…….. eq.2 ⇒ a = 12/b Now, using the value of a in eq.1, we get ⇒ (12/b)² – b² = - 7 ⇒ 144 – b⁴ = -7b² ⇒ b⁴ - 7b² - 144 = 0 Simplify and get the value of b², we get, ⇒ b² = - 9 or b² = 16 As b is real no. so, b² = 16 b = 4 or b = - 4 Therefore, a = 3 or a = - 3 Hence the square root of the complex no. is 3 + 4i and - 3 - 4i.

Answer»

Let, (a + ib)²= - 7 + 24i

Now using, (a + b)² = a² + b² + 2ab

a² + (bi)² + 2abi = -7 + 24i

Since i² = -1

a² - b² + 2abi = -7 + 24i

Now, separating real and complex parts, we get

⇒ a² - b² = - 7…………..eq.1

⇒ 2ab = 24…….. eq.2

⇒ a = 12/b

Now, using the value of a in eq.1, we get

⇒ (12/b)² – b² = - 7

⇒ 144 – b⁴ = -7b²

⇒ b⁴ - 7b² - 144 = 0

Simplify and get the value of b², we get,

⇒ b² = - 9 or b² = 16

As b is real no. so, b² = 16

b = 4 or b = - 4

Therefore, a = 3 or a = - 3

Hence the square root of the complex no. is 3 + 4i and - 3 - 4i.

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