`(1 + i)^5` Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form.

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`z=(1+i)^5`

`r=sqrt{1^2+1^2)=sqrt2`

`theta=arctan(1/1)=arctan(1)=pi/4`

`z=sqrt2(cos(pi/4)+isin(pi/4))`

DeMoivre's Theorem

`z^n=[r(costheta+isintheta)]^n=r^n[cosntheta+isinntheta]`

`z^5=[sqrt2(cos(pi/4)+isin(pi/4))]^5=(sqrt2)^5[cos5(pi/4)+isin5(pi/4)]`

`z^5=4sqrt2[cos((5pi)/4)+isin((5pi)/4)]`

`z^5=4sqrt2[-sqrt2/2+(-sqrt2/2)i]`

`z^5=-4-4i`

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`z=(1+i)^5`

`r=sqrt{1^2+1^2)=sqrt2`

`theta=arctan(1/1)=arctan(1)=pi/4`

`z=sqrt2(cos(pi/4)+isin(pi/4))`

DeMoivre's Theorem

`z^n=[r(costheta+isintheta)]^n=r^n[cosntheta+isinntheta]`

`z^5=[sqrt2(cos(pi/4)+isin(pi/4))]^5=(sqrt2)^5[cos5(pi/4)+isin5(pi/4)]`

`z^5=4sqrt2[cos((5pi)/4)+isin((5pi)/4)]`

`z^5=4sqrt2[-sqrt2/2+(-sqrt2/2)i]`

`z^5=-4-4i`

 

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