Question

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

Answer 1

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

Explanation:

Suppose `z = cos((2pi)/5) + i sin((2pi)/5)`

Then by De Moivre's formula we find:

`z^5 = cos(2pi)+i sin(2pi) = 1`

So `sqrt(z^5) = sqrt(1) = 1`

However:

`z^(5/2) = cos(pi) + i sin(pi) = -1`

So: `7(z^5)^(1/2) = 7 != -7 = 7z^(5/2)`