Evaluate √(1+4√-3)

Question

√(1+4√-3)

Answer 1

Let, (a + ib)² = 1 + 4√3 i

Now using, (a + b)² = a² + b² + 2ab

a² + (bi)² + 2abi = 1 + 4√3 i

Now, separating real and complex parts, we get

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

⇒ \((\frac{2\sqrt3}{b})^2\) – b² = 1

⇒ 12 – b⁴ = b²

⇒ b⁴ + b² - 12= 0

Simplify and get the value of b², we get,

b² = - 4 or b² = 3

As b is real no. so, b² = 3

b = √3 or b = -√3

Therefore, a = 2 or a = - 2

Hence the square root of the complex no. is 2 + √3 i and - 2 - √3 i.