`(-1+i)^6`

De Moivre's Theorem is used to compute the powers and roots of a complex number. The formula is:

`[r(costheta + isintheta)]^n=r^n(cos(nxxtheta) + isin(n xx theta))`

To be able to apply it, convert the complex number z=-1+i to trigonometric form.Take note that that the trigonometric form of

`z=x+yi`

is

`z=r(costheta + isintheta)`

where

`r =sqrt(x^2+y^2)` and `theta=tan^(-1) y/x`

Applying these two formulas, the values of r and theta of z=-1+i are:

`r=sqrt((-1)^2+1^2)=sqrt2`

`theta = tan^(-1) (1/(-1))=tan^(-1) (-1) = -45^o`

Since x is negative and y is positive, theta is located at the second quadrant. So the equivalent positive angle of theta is:

`theta =180^o +(-45^o)=135^o`

Then, plug-in the values of r and theta to the trigonometric form

`z=r(costheta + isintheta)`

`z=sqrt2(cos135^o +isin135^o)`

Now that z=-1+i is in trigonometric form, proceed to compute z^6 .

`z^6=(-1+i)^6`

`=[sqrt2(cos135^o + isin135^o)]^6`

`= (sqrt2)^6(cos(6xx135^o)+isin(6xx135^o)`

`=8(cos810^o + isin810^o)`

`=8(0 + 1i)`

`=8i`

**Therefore, `(-1+i)^6=8i` .**

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