How do you write 7sqrt(z^5) as an exponential expression?

Jan 14, 2016

If $z$ is a Complex number (as the choice of $z$ rather than $x$ might imply), then this can safely be written as $7 {\left({z}^{5}\right)}^{\frac{1}{2}}$, but this may or may not be equal to $7 {z}^{\frac{5}{2}}$

Explanation:

Suppose $z = \cos \left(\frac{2 \pi}{5}\right) + i \sin \left(\frac{2 \pi}{5}\right)$

Then by De Moivre's formula we find:

${z}^{5} = \cos \left(2 \pi\right) + i \sin \left(2 \pi\right) = 1$

So $\sqrt{{z}^{5}} = \sqrt{1} = 1$

However:

${z}^{\frac{5}{2}} = \cos \left(\pi\right) + i \sin \left(\pi\right) = - 1$

So: $7 {\left({z}^{5}\right)}^{\frac{1}{2}} = 7 \ne - 7 = 7 {z}^{\frac{5}{2}}$