Evaluate √(-2+2√3i)

Question

√(-2+2√3i)

Answer 1

Let, (a + ib)² = - 2 + 2√3 i

Now using, (a + b)² = a² + b² + 2ab

⇒ a² + (bi)² + 2abi = - 2 + 2√3 i

Since i² = -1

⇒ a² - b² + 2abi = - 2 + 2√3 i

Now, separating real and complex parts, we get

⇒ a² - b² = - 2…………..eq.1

⇒ 2ab = 2√3 ……..eq.2

⇒ a = √3/b

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

⇒ (√3/b)² – b² = -2

⇒3 – b⁴ = -2b²

⇒ b⁴ - 2b² - 3= 0

Simplify and get the value of b², we get,

⇒ b² = -1 or b² = 3

As b is real no. so, b² = 3

b = √3 or b = -√3

Therefore, a = 1 or a = -1

Hence the square root of the complex no. is 1 + √3 i and -1 - √3 i.