`(3 - 2i)^8` Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form.

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`(3-2i)^8`

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

`[ r(cos theta +isintheta)]^n = r^n(cos(ntheta) + isin(ntheta))`

Notice that its formula is in trigonometric form. So to compute `(3-2i)^8` , it is necessary to convert the complex...

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`(3-2i)^8`

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

`[ r(cos theta +isintheta)]^n = r^n(cos(ntheta) + isin(ntheta))`

Notice that its formula is in trigonometric form. So to compute `(3-2i)^8` , it is necessary to convert the complex number` z= 3-2i` to trigonometric form `z=r(cos theta+isin theta` ).

 To convert `z=x+yi`  to  `z=r(costheta +isintheta)` , apply the formula

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

So,

` r=sqrt(3^2+(-2)^2)=sqrt(9+4)=sqrt13`

 

`theta = tan^(-1) (-2)/3=-33.69007^o`

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

`theta =360^o +(-33.69007^o)=326.30993^o`

Hence, the trigonometric form of the complex number 

`z=3-2i`

is

`z=sqrt13(cos326.30993^o + isin326.30993^o)`

Now that it is in trigonometric form, proceed to apply the formula of De Moivre's Theorem to compute `z^8` .

`z^8=(3-2i)^8`

     `=[sqrt13(cos326.30993^o +isin326.30993^o)]^8`

     `=(sqrt13)^8(cos(8xx326.30993^o) +isin(8xx326.30993^o))`

     `=28561(cos(8xx326.30993^o) +isin(8xx326.30993^o))`

     `= -239+28560i`

 

Therefore, `(3-2i)^8=-239+28560i` .

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