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

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

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Answer 1 · 2016-01-14T22:46:00

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

Explanation:

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

Then by De Moivre's formula we find:

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

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

However:

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

So: $7 {(z^{5})}^{\frac{1}{2}} = 7 \neq - 7 = 7 z^{\frac{5}{2}}$ $7(z^5)^(1/2) = 7 != -7 = 7z^(5/2)$

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

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