problem

what is the magnitude of j^(5/8)? please give an explanation.

$\begin{array}{|rcll|} \hline j^{(\frac{5}{8})} = \sqrt[8]{j^5} = \sqrt[8]{j}= \sqrt[8]{z}\quad & | \quad j^5 = j^2\cdot j^2 \cdot j = (-1)\cdot(-1) \cdot j = 1\cdot j = j \\ \hline \end{array}$

Formula:

$\begin{array}{|lcll|} \hline z &=& a +bj \qquad |z| = \sqrt{a^2+b^2} \qquad |z| = \text{ magnitude of z } \\ \hline \end{array}$

$\begin{array}{|lcll|} \hline z &=& j \\ &=& 0+1\cdot j \qquad a = 0 \qquad b = 1 \\\\ |z| &=& \sqrt{0^2+1^2} \\ &=& \sqrt{1} \\ |z| &=& 1 \\ \hline \end{array}$

$\begin{array}{|rcll|} \hline \sqrt[8]{z} &=&\sqrt[8]{j} \\ &=& \sqrt[8]{|z|} \cdot \left( \cos( \frac{\varphi}{8} ) + i\cdot \sin(\frac{\varphi}{8}) \right) \\ &=& \sqrt[8]{1} \cdot \left( \cos( \frac{\varphi}{8} ) + i\cdot \sin(\frac{\varphi}{8}) \right) \\ &=& 1 \cdot \left( \cos( \frac{\varphi}{8} ) + i\cdot \sin(\frac{\varphi}{8}) \right) \\ \hline \end{array}$

The magnitude of $j^{\frac{5}{8}} = j^{\frac{1}{8}} = \sqrt[8]{j}$ is:

$\begin{array}{rcll} \sqrt[8]{|z|} &=& \sqrt[8]{1} \\ &=& 1 \\ \end{array}$