Simple Roots

Given, $\Large x\in \mathbb{R}$ . Now, the square root function is a bijection from $\mathbb{R_0^+}$ to $\mathbb{R_0^+}$ , i.e., the square root function generally returns positive value (including zero).

Now, if $x$ is (+ve), then the value of $\sqrt{x^2}=x$ .

If $x$ is (-ve), then the value of $\sqrt{x^2}=(-x)$ .

These two values can be collectively expressed by saying that $\sqrt{x^2}=|x|$ .

Now, coming onto $\Large \sqrt[3]{x^3}$ , for all $\Large x\in \mathbb{R}$ , we have $\large \sqrt[3]{-x^3} = \sqrt[3]{(-1)} \times \sqrt[3]{x^3} = (-1) \times x = (-x)$ .

So, the answers are $|x|$ and $\Large (-x)$ respectively.

P.S. -- Sorry for the bad formatting! I am a little bad with LaTeX. :P

Square root of a number could be +tive or - tive. Hence absolute value wound take care of both values. While cube root will have the same sign as that of the number, - tive here.