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

lfryerda | High School Teacher | (Level 2) Educator

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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

`(x-2)(x^2+2x+4)=0`

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

`x={-2+-sqrt{4-4(4)}}/2`

`=-1+-sqrt{1-4}`

`=-1+-isqrt3`

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}`

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