Question

find square root of 3 + 2√10i​

Answer 1

Answer:

√5 + √2i or -√5 - √2i

Step-by-step explanation:

Let us assume that square root of 3 + 2√10i

Can be written as a + bi

Now,

=> 3 + 2√10i = a + bi

On squaring both sides

=> (3 + 2√10i) = (a + bi) ²

=> a² + (bi) ² + 2abi = 3 + 2√10i

=> a² + b²(-1) + 2ab(i) = 3 + 2√10i

=> (a² - b²) + 2ab(i) = 3 + 2√10i

Equating both real and imaginary parts

=> a² - b² = 3 - - - (1)

=> 2ab = 2√10

=> ab = √10 - - - (2)

=> b = √10/a

Putting the value of b

=> a² - (√10/a)² = 3

=> a² - 10/a² = 3

=> a⁴ - 3a² - 10 = 0

=> a⁴ - 5a² + 2a² - 10 = 0

=> a²(a² - 5) + 2(a² - 5) = 0

=> (a² - 5)(a² + 2) = 0

But a is real so

a = √5

=> b = √10/√5

=> b = √2

So square root of 3 + 2√10i is

Either

√5 + √2i or -√5 - √2i