dertermine the three cubic roots 8. give the answer in the form a+bj. also give the main root in the exponential form.  

Expert Answers
lfryerda eNotes educator| Certified Educator

The cubic roots of a number is equivalent to solving the equation `x^3=8` which can be factored into:

`x^3-8=0`     use division to get


Now the first factor has solution `x=2` , and the second factor can have solutions using the quadratic formula




This means that the three roots are `x=2,-1+-isqrt3` .

In exponential form, a complex number is given as `r e^{i theta}` where r is the absolute value of the number and `theta` is the angle formed from the positive real axis.  The angle is found by `theta=tan^{-1}(y/x)` where y is the imaginary value and x is the real value.  In this case, we see that we have special triangles for the two complex roots, and using CAST, we get:

The roots in exponential form are `2,2e^{{2i pi}/3}, 2e^{{4i pi}/3}`