√(3+4√-7)

1.	√(3+4√-7)
Answer» Let, (a + ib)² = 3 + 4√7 i Now using, (a + b)² = a² + b² + 2ab ⇒ a² + (bi)² + 2abi = 3 + 4√7 i Since i² = -1 ⇒ a² - b² + 2abi = 3 + 4√7 i Now, separating real and complex parts, we get ⇒ a² - b² = 3 …………..eq.1 ⇒ 2ab = 4√7 …….. eq.2 ⇒ a = 2√7/b Now, using the value of a in eq.1, we get ⇒ \((\frac{2\sqrt7}{b})^2\) – b² = 3 ⇒ 12 – b⁴ = 3b² ⇒ b⁴ + 3b² - 28 = 0 Simplify and get the value of b², we get, ⇒ - b² = -7 or b² = 4 as b is real no. so, b² = 4 b = 2 or b = - 2 Therefore, a = √7 or a = -√7 Hence the square root of the complex no. is √7 + 2i and -√7 - 2i.

Answer»

Let, (a + ib)² = 3 + 4√7 i

Now using, (a + b)² = a² + b² + 2ab

⇒ a² + (bi)² + 2abi = 3 + 4√7 i

Since i² = -1

⇒ a² - b² + 2abi = 3 + 4√7 i

Now, separating real and complex parts, we get

⇒ a² - b² = 3 …………..eq.1

⇒ 2ab = 4√7 …….. eq.2

⇒ a = 2√7/b

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

⇒ \((\frac{2\sqrt7}{b})^2\) – b² = 3

⇒ 12 – b⁴ = 3b²

⇒ b⁴ + 3b² - 28 = 0

Simplify and get the value of b², we get,

⇒ - b² = -7 or b² = 4

as b is real no. so, b² = 4

b = 2 or b = - 2

Therefore, a = √7 or a = -√7

Hence the square root of the complex no. is √7 + 2i and -√7 - 2i.

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