+185 The number \(16\) has four fourth roots. In other words, there are four complex numbers that can be entered in the square in the equation below: \(\square^4=16.\) Find them.
I know that 2 is one, but I don't know how to find the others. Could anyone please provide an explanation with their answer? Thank you!
+738 hi otterstar!
so we know that \((-2)^4=16\) since -2*-2*-2*-2=16
since there are 4 roots, and we've already covered all the real ones, we have to look for complex numbers now.
we also know that \(i^4=1\), so that means that \((2i)^4=16\) since 2^4 already equals 16, and multiplying it by 1 will equal 16 too.
lastly, we know that \((-2i)^4=16\) since we already know that -2^4 equals 16 (as we found earlier) and multiplying it by i^4=1 will equal 16 too.
so the solutions are \(\boxed{2,-2,2i,-2i}\)
i hope this helped you! :)
