Question

Please Help!! What are the fourth roots of -64??

Answer 1

do you want all 4 roots, including the complex roots?

Answer 2

Yes, please!

Answer 3

one way is write this in polar coordinates
the fourth root of -64 is
\[ \left(64 e^{i\pi}\right)^{\frac{1}{4}} = (64)^{\frac{1}{4}} e^{\frac{i\pi}{4}} \]

Answer 4

the roots are spaced equally around the unit circle

Answer 5

I am still unsure of this.. the possible answers are in the form: r(cos theta + i sin theta)

Answer 6

does this make sense?

Answer 7

What the hell. I am unsure how to find the such a root of a number as it is.

Answer 8

remember that
\[ r e^{i\theta} = r(\cos(\theta)+i \sin(\theta)) \]

Answer 9

64 is 8*8 and sqrt(8) is 2sqrt(2) This will be the magnitude
the first root will be at \(\theta= \frac{\pi}{4} \)
this is written either as
\[ 2 \sqrt{2} e^{i\frac{\pi}{4}} = 2 \sqrt{2} \left( \cos(\frac{\pi}{4})+i \sin(\frac{\pi}{4})\right)\]

the 4 roots are spaced equally around \(2\pi\), so their spacing is
\[\frac{2\pi}{4}= \frac{\pi}{2}\]
so the next root will be at \(\theta= \frac{\pi}{4}+ \frac{\pi}{2} = \frac{3\pi}{4}\)
keep adding pi/2 until you get all 4 roots