Question

write (square root) of -32 in simplified complex form

Answer 1

Hey there :)
Here is a clever way that we can break up the negative sign and the 32.\[\large\rm \sqrt{-32}\quad=\sqrt{-1\cdot 32}\quad=\sqrt{-1}\cdot\sqrt{32}\]

Answer 2

Recall that we define our `imaginary unit` in this way: \(\large\rm \color{orangered}{\sqrt{-1}=i}\)

Answer 3

Do you understand how that is going to help us?\[\large\rm \color{orangered}{\sqrt{-1}}\cdot\sqrt{32}\quad=\quad ?\]

Answer 4

yes

Answer 5

\[\large\rm \color{orangered}{\sqrt{-1}}\cdot\sqrt{32}\quad=\color{orangered}{i}\cdot\sqrt{32}\]Good, we can input our imaginary unit, ya?
Another good step we can apply,
is to pull anything out the root if we can.

Answer 6

So does the value 32 contain any `factors` that are perfect squares?
These are perfect squares: 4, 9, 16, 25, 36, ...

Does 32 have any of those values as factors?

Answer 7

ya 16 and 2

Answer 8

Ok great, let's apply that earlier trick again,\[\large\rm i\cdot \sqrt{32}\quad=i\cdot \sqrt{16\cdot2}\quad=i\cdot \sqrt{16}\cdot\sqrt{2}\]

Answer 9

And then what's the last simplification step you can do? :)

Answer 10

See it?

Answer 11

square root 16

Answer 12

i