# What is the difference between: $\sqrt {z^3}$ and: $4 \sqrt {z^3}$

I was working on math and I had to solve this problem:

Rewrite the expression in radical form: $z^\frac{3}{4}$

My first attempt was wrong: $4 \sqrt {z^3}$
I was confused as to why this was wrong until I saw the difference between the two solutions. This got me thinking: why are they different? (and in fact, I don't know how to say: $\sqrt {z^3}$)
If you wouldn't mind, because I don't know what it's called, could you please explain the differences between the two solutions? (and maybe how to say the former ;) )

Thanks for you time!

• is $z$ a complex number? Dec 21, 2016 at 15:40
• First one is the forth root of $z^3$ and the second one is four times the square root of $z^3$. Dec 21, 2016 at 15:42
• @Dr.SonnhardGraubner no, $z$ is not a complex number. Dec 21, 2016 at 15:43

$\sqrt {z^3}=z^{3/4}$ and $4 \sqrt {z^3}=4 \times z^{3/2}.$ i.e. the former is the fourth root of $z^3$ and the latter is multiplying the square root of $z^3$ by $4.$
it is $$4\sqrt{z^3}=4z^{3/2}$$ and $$\sqrt{z^3}=z^{3/4}$$