How to write such $\sqrt{\sqrt{x}}$ in equation?

Question

Is it correct to write $\sqrt{\sqrt{x}}$ as equation/formula? or $\sqrt[4]{x}$?

I mean, for example, if $x = 10000$, the outcome will be $10$.

Answer 1

It's not wrong to write $\sqrt{\sqrt{x}}$ in the same way that it's not wrong to write $\sqrt{\sqrt{\frac{2x^1}{2}}}$, but it's generally preferable to write expressions as simply as possible.

The phrase "square root" means the same thing as "one-half power". So,

$$ \sqrt{\sqrt{x}} = \sqrt{x^{1/2}} = (x^{1/2})^{1/2} = x^{1/2 \cdot 1/2} = x^{1/4}. $$

For the same reason as with square root, it would be appropriate to write either $x^{1/4}$ or $\sqrt[4] x$.