$x^{1/2}-4x^{1/4}=0 \Rightarrow x = 0, 256$ – cannot arrive at this.

Question

I am asked to solve:

$x^\frac{1}{2}-4x^\frac{1}{4}=0$

The solution provided is x=0 and 256. I get stuck at $x^2-4x=0$:

$x^\frac{1}{2}-4x^\frac{1}{4}=0$

$(x^\frac{1}{2}-4x^\frac{1}{4})^\frac{4}{1}$ # raise both sides to $\frac{4}{1}$ as the reciprocal of $\frac{1}{4}$

$x^2-4x=0$ # where to go from here?

Perhaps my previous step was incorrect? For the first term in brackets, $x^\frac{1}{2}$ can be written $\sqrt{x}$ and then I'm raising to the power of 4 so $x^2$ since $(\sqrt{x})^2$ is just $x$ but I'm raising to power of 4 not 2 so $x^2$ for the first component.

For the second component since I'm raising to the reciprocal it cancels out to just $-4x$

Where did I go wrong and how can I arrive at solutions 0 and 256?

Answer 1

$\sqrt[4]x=y,x=y^4$

$0=y^2-4y=y(y-4)$

$4^4=(4^2)^2=?$