Simple question on complex numbers and multiplying by the conjugate

Question

So I'm starting off with complex numbers and am unsure about the following:

I want to evaluate $\frac{\sqrt{-7}}{\sqrt{-9}\sqrt{-1}}$ and write my answer in the form $a+bi$. So after some evaluation I got this equal to $ - \frac{\sqrt{7}}{3} i$. So there's no real part i.e. $a = 0$ and $b = - \frac{\sqrt{7}}{3}$. Is this right and if so, why don't I need to use the conjugate at all?

Also, if for example I want to write $-5 (\frac{i}{2})$ in the form $a +bi$, is this just $\frac{-5}{2} i$?

If so, then when exactly do I use the conjugate? Only for division of two complex numbers where both top and both have a real and imaginary part?

Thanks

Answer 1

You use the conjugate in the following cases:

$\cfrac{7+\sqrt3}{7-\sqrt3}$ (conjugate is $7+\sqrt3$)

$\cfrac{10+i}{10-i}$ (conjugate is $10+i$)

$\cfrac{\sqrt{10}-3i}{\sqrt7+4i}$ (conjugate is $\sqrt7-4i$)

Note: There is addition or subtraction in the denominator, with multiplication or division, you don't need to use the conjugate.

Example:

$\cfrac{10}{\sqrt{5}}$

Although the conjugate above is $-\sqrt5$, you can simply multiply the numerator and denominator by $\sqrt5$ to rationalize the exponent.