$\sqrt[3]{4\frac{12}{125}} = x$

Concept & Strategy To find the root of a mixed fraction, you must first convert it into an improper fraction. Once converted, apply the cube root separately to the numerator and the denominator using standard radical properties. Step-by-Step Solution * **Given expression**: $$\sqrt[3]{4\frac{12}{125}}$$ * Convert the mixed fraction $4\frac{12}{125}$ to an improper fraction: $$\text{Numerator} = (4 \times 125) + 12 = 500 + 12 = 512$$ $$\text{Fraction} = \frac{512}{125}$$ * Apply the cube root to the new improper fraction: $$\sqrt[3]{\frac{512}{125}} = \frac{\sqrt[3]{512}}{\sqrt[3]{125}}$$ * Evaluate the cube roots of these known perfect cubes: $$\sqrt[3]{512} = 8$$ $$\sqrt[3]{125} = 5$$ * The resulting fraction is $\frac{8}{5}$. Convert this back to a mixed fraction for the final answer: $$\frac{8}{5} = 1\frac{3}{5}$$ Exam Strategy & Shortcut Memorize standard cubes up to $10$. Once you see the denominator is $125$, you know its cube root is $5$. The numerator becomes $512$ ($4 \times 125 + 12$). Since $\sqrt[3]{512}$ is $8$, the fraction $8/5$ quickly converts to $1\frac{3}{5}$ in your head. Common Pitfall Attempting to take the cube root of the whole number ($4$) and the fraction piece ($\frac{12}{125}$) separately. This mathematically violates radical rules and leads to nonsensical dead ends. Always convert to improper fractions first. Final Answer **Therefore, the correct answer is $1\frac{3}{5}$.**