1. **State the problem:** We are given the function $$y = 3 \sqrt[3]{x^5}$$ and want to understand its shape and behavior. 2. **Rewrite the function using exponent rules:** The cube root of $$x^5$$ can be written as $$x^{\frac{5}{3}}$$. So the function becomes $$y = 3x^{\frac{5}{3}}$$. 3. **Important rules:** - The exponent $$\frac{5}{3}$$ means the fifth power of $$x$$ is taken first, then the cube root. - Since the root is odd (cube root), the function is defined for all real $$x$$, including negative values. 4. **Analyze the function:** - For $$x > 0$$, $$y$$ increases as $$x$$ increases because $$x^{\frac{5}{3}}$$ grows. - For $$x < 0$$, since the cube root of a negative number is negative, $$y$$ will be negative but still defined. - The coefficient 3 stretches the graph vertically by a factor of 3. 5. **Graph shape:** - The graph passes through the origin (0,0). - It is increasing for all $$x$$. - The curve is steeper than $$y = x^{\frac{1}{3}}$$ because of the higher exponent. **Final function:** $$y = 3x^{\frac{5}{3}}$$