1. **State the problem:** We want to analyze the function $$y = \sqrt[3]{x^5}$$. 2. **Rewrite the function using exponents:** The cube root can be expressed as a fractional exponent: $$y = (x^5)^{\frac{1}{3}} = x^{\frac{5}{3}}$$. 3. **Important rules:** - When raising a power to another power, multiply the exponents. - The domain of $$y = x^{\frac{5}{3}}$$ includes all real numbers because the cube root is defined for negative and positive values. 4. **Simplify and analyze:** - The function is increasing for all real $$x$$. - For $$x > 0$$, $$y$$ is positive. - For $$x < 0$$, since $$5$$ is odd, $$x^5$$ is negative, and the cube root of a negative number is negative, so $$y$$ is negative. 5. **Final expression:** $$y = x^{\frac{5}{3}}$$. This function is continuous and smooth for all real $$x$$, with no restrictions on the domain.