Find the fifth roots of 32(cos 280° + i sin 280°).

ANSWER[tex]2 ( { \cos \: 56 \degree + i \sin \:56 \degree) }[/tex]EXPLANATIONThe complex number given to us is in the polar form,32(cos 280° + i sin 280°)The fifth root is [tex] {32}^{ \frac{1}{5} } ( { \cos280 \degree + i \sin280 \degree) }^{ \frac{1}{5} } [/tex]This is equal to:[tex]2 ( { \cos280 \degree + i \sin280 \degree) }^{ \frac{1}{5} } [/tex]According to the DeMoivre's Theorem,[tex]( { \cos \theta \: \degree + i \sin\theta \degree) }^{ \frac{p}{q} } = ( { \cos \frac{p}{q} \theta \degree + i \sin \frac{p}{q} \theta \degree) }[/tex]We now use the DeMoivre's Theorem to obtain:[tex]2 ( { \cos280 \degree + i \sin280 \degree) }^{ \frac{1}{5} } = 2 ( { \cos \: \frac{1}{5} \times 280 \degree + i \sin \:\frac{1}{5} \times 280 \degree) }[/tex][tex]2 ( { \cos280 \degree + i \sin280 \degree) }^{ \frac{1}{5} } = 2 ( { \cos \: 56 \degree + i \sin \:56 \degree) }[/tex]