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A cube root is a special mathematical operation that tells us what number, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because 2 multiplied by itself twice (2 × 2 × 2) equals 8. When expressing this in mathematical terms, we often use the notation \sqrt[3]{...}.

If we take the cube root of an expression like \sqrt[3]{x^3}, we are essentially undoing the operation of cubing the number. Here’s a simpler way to understand it: Raising a number to the power of 3 and then taking the cube root will give us back the original number, which can be expressed as: \[ \sqrt[3]{x^3} = x. \] This is because \[(x^3)^{1/3} = x^{3 \cdot 1/3} = x.\]

Applying a negative sign to an expression changes the sign of the result from positive to negative or vice versa. In our exercise, we began with the expression \- \sqrt[3]{x^{3}}. After finding the cube root in the first step, we were left with \sqrt[3]{x^3} = x.

The final step is to apply the negative sign. This means we simply place the negative sign in front of x, giving us -x. It’s important to note that the negative sign is not part of the cube root operation itself but is applied to the result afterward.
This transformation can be visually understood as follows:

\[\text{Original expression: } -\sqrt[3]{x^3}\]
\[\text{Cube root result: } x\]
\[\text{Apply negative sign: } -x\]

Simplifying expressions involves breaking down a complex mathematical expression into simpler, more manageable parts. In our given problem, we simplified the expression \- \sqrt[3]{x^{3}}.

Here’s a step-by-step overview of how we simplified it:

• First, we identified that we needed to take the cube root of \(x^{3}\).
• Second, we computed the cube root, realizing that the cube root of \(x^{3}\) is simply \(x\).
• Finally, we applied the negative sign, transforming the expression into its simplest form, \-x.

Simplifying expressions is an essential skill in mathematics, making it easier to solve equations and understand the underlying concepts. By following these steps, we ensure that our mathematical work is both accurate and efficient. Simplifying complex expressions step-by-step can boost your confidence in tackling more challenging problems in the future.