Evaluate √(7-30√-2)

Question

√(7-30√-2)

Answer 1

Let, (a + ib)² = 7 - 30√2 i

Now using, (a + b)² = a² + b² + 2ab

⇒ a² + (bi)² + 2abi = 7 - 30√2 i

Since i² = -1

⇒ a² - b² + 2abi = 7 - 30√2 i

Now, separating real and complex parts, we get

⇒ a² - b² = 7 …………..eq.1

⇒ 2ab = 30√2 …….. eq.2

⇒ a = \(\frac{15\sqrt2}{b}\)

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

⇒ \((\frac{15\sqrt2}{b})^2\) – b² = 7

⇒ 450 – b⁴ = 7b²

⇒ b⁴+ 7b² - 450 = 0

Simplify and get the value of b², we get,

⇒ b² = -25 or b² = 18

As b is real no. so, b² = 18

b = 3√2 or b = -3√2

Therefore, a = 5 or a = - 5

Hence the square root of the complex no. is 5 + 3√2 i and - 5 - 3√2 i.