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Exercise 23 Show that for $p$ and $q$ relatively prime, $\alpha^{p/q}$ has $q$ roots, of the form $\alpha^{p/q} = r^{p/q} \exp \left[ip(\theta + 2n\pi)/q\right]$ $= r^{p/q} \left\{\cos \left[p(\theta + 2n\pi)/q\right] + i \sin \left[p(\theta + 2n\pi)/q\right]\right\}$, for $n = 0, 1, \dots, q - 1$. Exercise 24 Use Eqn. 31 to compute all the rational roots listed below in polar form: a. $\sqrt{2}i$, b. $i^{1/3}$, c. $1^{2/3}$, d. $(1 + i)^{1/4}$, e. $(-4)^{2/5}$.

          Exercise 23 Show that for $p$ and $q$ relatively prime, $\alpha^{p/q}$ has $q$ roots, of the form
$\alpha^{p/q} = r^{p/q} \exp \left[ip(\theta + 2n\pi)/q\right]$
$= r^{p/q} \left\{\cos \left[p(\theta + 2n\pi)/q\right] + i \sin \left[p(\theta + 2n\pi)/q\right]\right\}$,
for $n = 0, 1, \dots, q - 1$.
Exercise 24 Use Eqn. 31 to compute all the rational roots listed below in polar form:
a. $\sqrt{2}i$,
b. $i^{1/3}$,
c. $1^{2/3}$,
d. $(1 + i)^{1/4}$,
e. $(-4)^{2/5}$.

$Exercise 23 Show that for p and q relatively prime, α^p/q has q roots, of the form α^p/q = r^p/q[ip(θ + 2nπ)/q] = r^p/q{cos[p(θ + 2nπ)/q] + i sin[p(θ + 2nπ)/q]}, for n = 0, 1, …, q - 1. Exercise 24 Use Eqn. 31 to compute all the rational roots listed below in polar form: a. √(2)i, b. i^1/3, c. 1^2/3, d. (1 + i)^1/4, e. (-4)^2/5.$

Added by Rachel A.

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