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: $\sqrt [4]{z^3}$)
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!

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

2 Answers 2


$\sqrt [4]{z^3}=z^{3/4}$ 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 $4.$

  • $\begingroup$ Awesome, thanks for the information! $\endgroup$ Dec 21, 2016 at 15:48

it is $$4\sqrt{z^3}=4z^{3/2}$$ and $$\sqrt[4]{z^3}=z^{3/4}$$


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