1. State the problem: Simplify $|x|^3$. 2. Interpretation: I read this as the cube of the absolute value, which is $|x|^3$. 3. Property: We have $|x|^3=(|x|)^3$ and also $|x^3|=|x|^3$, so examining sign cases will give the result. 4. Case 1 -- nonnegative $x$: If $x\ge 0$ then $|x|=x$ and therefore $$|x|^3 = x^3$$. 5. Case 2 -- negative $x$: If $x<0$ then $|x|=-x$ and therefore $$|x|^3 = (-x)^3 = -x^3$$. 6. Final answer: Combining both cases, $|x|^3 = x^3$ for $x\ge 0$ and $|x|^3 = -x^3$ for $x<0$.