To simplify the expression $\sqrt[3]{x^{15}}$, we follow the steps below:

The cube root of $x^{15}$ is the same as raising $x^{15}$ to the power of $\frac{1}{3}$, which gives:

$\sqrt[3]{x^{15}} = x^{\frac{15}{3}} = x^5$

Thus, the simplified expression is:

$x^5$

Explanation:

• The cube root of $x^{15}$ can be written as $(x^{15})^{\frac{1}{3}}$.
• Using the rule $(a^m)^n = a^{m \cdot n}$, we get $x^{15 \cdot \frac{1}{3}} = x^5$.

Further questions for exploration:

1. What are the general rules for simplifying radical expressions?
2. How can we simplify square roots or fourth roots in similar ways?
3. What happens if the exponent is negative, such as $\sqrt[3]{x^{-15}}$?
4. Can we simplify expressions like $\sqrt[3]{x^n}$ where $n$ is any integer?
5. How do different values of $x$ (like rational or irrational) affect the simplification process?

Tip:

Always simplify exponents first, and remember that the cube root of $x^n$ simplifies to $x^{\frac{n}{3}}$.