If you have a problem like $$\frac{3}{\sqrt{-9}}$$does it matter if you bring the i out first or rationalize first? I believe that it doesn't matter which because it'll come out as i?

Question

phi · Answer

it does not matter, but generally people write 
$$ \sqrt{-9} \text{  as  }i\sqrt{9} $$
so
$$ \frac{\cancel{3}}{\cancel{3}i}= \frac{i}{i*i}= -i $$

anonymous · Answer

Wait, if you do it the other way, wouldn't it become i?
$$\frac{3}{\sqrt{-9}} \times \frac{\sqrt{-9}}{\sqrt{-9}} \implies \frac{9i}{9} \implies i$$Or did I do something wrong?

phi · Answer

sqrt(-9)*sqrt(-9)= -9

anonymous · Answer

Ohh...I thought that you would multiply was was inside before simplify the radical...I guess I was wrong. THank you :)

phi · Answer

I think
$$ \sqrt{a \cdot b} =\sqrt{a}\cdot \sqrt{b} $$
only works for positive a,b

anonymous · Answer

Oh ok. THat makes sense since you can't really have two answers for something like this. Again, thank you. I'll double check the rule in a second.

anonymous · Answer

Yup. THat is the case. Would that also apply to dividing radicals?