1. **State the problem:** Simplify the expression $$2x \sqrt[3]{16x^4} + \sqrt[3]{375x^8} - 2x^2 \sqrt[3]{54x}$$. 2. **Recall the cube root and exponent rules:** - $\sqrt[3]{a^m} = a^{\frac{m}{3}}$. - When multiplying powers with the same base, add exponents: $a^m \cdot a^n = a^{m+n}$. - When factoring inside cube roots, separate perfect cubes. 3. **Simplify each cube root:** - $\sqrt[3]{16x^4} = \sqrt[3]{16} \cdot \sqrt[3]{x^4} = \sqrt[3]{2^4} \cdot x^{\frac{4}{3}} = 2 \sqrt[3]{2} \cdot x^{\frac{4}{3}}$ because $16 = 2^4$ and $\sqrt[3]{2^3} = 2$. - $\sqrt[3]{375x^8} = \sqrt[3]{375} \cdot \sqrt[3]{x^8} = \sqrt[3]{125 \cdot 3} \cdot x^{\frac{8}{3}} = 5 \sqrt[3]{3} \cdot x^{\frac{8}{3}}$ because $125 = 5^3$. - $\sqrt[3]{54x} = \sqrt[3]{54} \cdot \sqrt[3]{x} = \sqrt[3]{27 \cdot 2} \cdot x^{\frac{1}{3}} = 3 \sqrt[3]{2} \cdot x^{\frac{1}{3}}$ because $27 = 3^3$. 4. **Rewrite the original expression substituting these:** $$2x \cdot 2 \sqrt[3]{2} \cdot x^{\frac{4}{3}} + 5 \sqrt[3]{3} \cdot x^{\frac{8}{3}} - 2x^2 \cdot 3 \sqrt[3]{2} \cdot x^{\frac{1}{3}}$$ 5. **Simplify coefficients and combine like terms:** - First term: $2x \cdot 2 \sqrt[3]{2} \cdot x^{\frac{4}{3}} = 4 \sqrt[3]{2} \cdot x^{1 + \frac{4}{3}} = 4 \sqrt[3]{2} \cdot x^{\frac{7}{3}}$ - Second term: $5 \sqrt[3]{3} \cdot x^{\frac{8}{3}}$ - Third term: $-2x^2 \cdot 3 \sqrt[3]{2} \cdot x^{\frac{1}{3}} = -6 \sqrt[3]{2} \cdot x^{2 + \frac{1}{3}} = -6 \sqrt[3]{2} \cdot x^{\frac{7}{3}}$ 6. **Group terms with common factors:** $$4 \sqrt[3]{2} x^{\frac{7}{3}} - 6 \sqrt[3]{2} x^{\frac{7}{3}} + 5 \sqrt[3]{3} x^{\frac{8}{3}}$$ 7. **Combine like terms:** $$ (4 - 6) \sqrt[3]{2} x^{\frac{7}{3}} + 5 \sqrt[3]{3} x^{\frac{8}{3}} = -2 \sqrt[3]{2} x^{\frac{7}{3}} + 5 \sqrt[3]{3} x^{\frac{8}{3}}$$ **Final simplified expression:** $$-2 x^{\frac{7}{3}} \sqrt[3]{2} + 5 x^{\frac{8}{3}} \sqrt[3]{3}$$