Write sqrt(-16+31i) as a complex number. Can you give me some hints, or if you give me the answer, please tell me the steps.

In exponential form, -16 + 31i = 34.89*e^(2.04726*pi*i).

Taking the square root, we get 5.91*e^(1.024*pi*i).

Converting this back to rectangular form gives 2 + 7i.

Let $x = \sqrt{-16 + 31i}$ and be in the form a + bi.

We have: $(a+bi)^2 = (a+bi)(a+bi) = a^2 + 2abi - b^2$

We know that $2abi = 31$, meaning $ab = {31 \over 2}$

Now, remember that $a^2 - b^2 = -16$

This gives us the system: 

$a^2 - b^2 = -16$          (i)

$ab = {31 \over 2}$                       (ii)

Using WA, we find the answer to be: $\sqrt{-16 + 31i} = \color{brown}\boxed{\sqrt{{\sqrt{1217}\over 2} - 8} + \sqrt{8 + {\sqrt{1217}\over 2}}i}$