`(-1 + i)^6` Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form.

`(-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...

`(-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` .