algebra precalculus - How to calculate $sqrt{frac{-3}{4} - i}$

I know that the answer to $\sqrt{\dfrac{-3}{4} - i}$ is $\dfrac12 - i$. But how do I calculate it mathematically if I don't have access to a calculator?

Answer

One of the standard strategies (the other strategy is to do what JM suggested in the comment to the qn) is to complete the square and use the fact that $i^2 = -1$.

$$\sqrt{\frac{-3}{4}-i}$$

Add and subtract 1 to get:

$$\sqrt{\frac{-3}{4}+1-i-1}$$

Use $i^2 = -1$ to get:

$$\sqrt{\frac{-3}{4}+1-i+i^2}$$

Simplify $\frac{-3}{4}+1$ to get:

$$\sqrt{\frac{1}{4}-i+i^2}$$

Rewrite $-i$ as $-2 \frac{1}{2}i$ to get:

$$\sqrt{\frac{1}{2^2}-2 \frac{1}{2}i+i^2}$$

Complete the square to get:

$$\sqrt{(\frac{1}{2}-i)^2}$$

Get rid of the square root to get:

$$\frac{1}{2}-i$$