Hi Gibsonj338
What are the five answers to this equations? Please show how you got to your answers.
\(x^5=243\)
3 is the first obvious solution.
Now, there will be 5 roots altogether.
each of these will be 3 units from the origin of the real/imaginary plane and they will be spaced at 2pi/5 radians intervals
\(\mbox{The 5 roots in polar form are:}\\~\\ 3,\; 3e^{2\pi i/5},\;3e^{4\pi i/5},\;3e^{6\pi i/5},\;3e^{8\pi i/5}\\~\\ 3,\;\; 3\left[cos(\frac{2\pi}{5})+isin(\frac{2\pi}{5})\right] ,\;\; 3\left[cos(\frac{4\pi}{5})+isin(\frac{4\pi}{5})\right] ,\;\; 3\left[cos(\frac{6\pi}{5})+isin(\frac{6\pi}{5})\right] ,\;\; 3\left[cos(\frac{8\pi}{5})+isin(\frac{8\pi}{5})\right]\\~\\ 3,\;\; 3\left[cos(72^0)+isin(72^0)\right] ,\;\; 3\left[cos(144^0)+isin(144^0)\right] ,\;\; 3\left[cos(216^0)+isin(216^0)\right] ,\;\; 3\left[cos(288^0)+isin(288^0)\right]\\~\\ 3,\;\; 3\left[0.3090+0.9511i\right] ,\;\; 3\left[-0.8090+0.5878i\right] ,\;\; 3\left[-0.9090-0.5878i\right] ,\;\; 3\left[0.3090-0.9511i\right]\\~\\ \mbox{So the 5 roots are:}\\~\\ 3,\;\;\quad 0.3090+2.8532i ,\;\; \:\quad -2.4271+1.7634i ,\;\; \;\quad -2.4271-1.7634i ,\;\; \;\quad 0.9271-2.8532i\\~\\\)
These are the approximations :)
The last guest answer gives the exact values.
More info and graph here.
http://www.wolframalpha.com/input/?i=x%5E5%3D243