1. **State the problem:** We want to simplify and understand the function $$y = \left(4 \ln(x^3)\right)^{\frac{1}{4}}$$. 2. **Recall the logarithm rule:** $$\ln(x^3) = 3 \ln(x)$$. This is because the logarithm of a power is the exponent times the logarithm of the base. 3. **Substitute the logarithm:** Replace $$\ln(x^3)$$ with $$3 \ln(x)$$ in the original function: $$y = \left(4 \times 3 \ln(x)\right)^{\frac{1}{4}} = \left(12 \ln(x)\right)^{\frac{1}{4}}$$ 4. **Rewrite the function:** The function is now $$y = \left(12 \ln(x)\right)^{\frac{1}{4}} = \sqrt[4]{12 \ln(x)}$$ 5. **Domain considerations:** Since $$\ln(x)$$ is defined only for $$x > 0$$, and the fourth root requires the inside to be non-negative, we need $$12 \ln(x) \geq 0$$ which means $$\ln(x) \geq 0$$ or $$x \geq 1$$. 6. **Final simplified function:** $$y = \sqrt[4]{12 \ln(x)}$$ for $$x \geq 1$$. This function grows slowly as $$x$$ increases beyond 1 because the logarithm grows slowly and the fourth root further moderates the growth.