Question

Find the roots of $(4+4i{)}^{\frac{p}{q}}$ . Where p and q are integers and q $\ne$ 0

• A. Where n = 0,1,2,3,..........q-1
• B. Where n = 0,1,2,3,..........q-1
• C. Where n = 0,1,2,3,..........q-1
• D. Where n = 0,1,2,3,..........q-1

Answer 1

The correct option is D

Where n = 0,1,2,3,..........q-1

Let z = 4+4i

$\left|z\right|$ = 4 $\sqrt{2}$

arg(z) = $ta{n}^{-1}$ $\left(\frac{4}{4}\right)$ = $\frac{\pi }{4}$

Express of Z in polar form

z = 4$\sqrt{2}$ $(cos\frac{\pi }{4}+isin\frac{\pi }{4})$
$z^{\frac{p}{q}}$ = ${\left(4\sqrt{2}\right)}^{\frac{p}{q}}$ ${[cos\frac{\pi }{4}+isin\frac{\pi }{4}]}^{\frac{p}{q}}$

= $(4\sqrt{2}{)}^{\frac{p}{q}}$ $\left[cos\frac{p}{q}\right(2n\pi +\frac{\pi }{4})+isin\frac{p}{q}(2n\pi +\frac{\pi }{4})]$

Where n = 0,1,2,3,..........q-1