1. **Problem:** Simplify the expression $$\sqrt[3]{8x^7}$$. 2. **Recall the rules:** - The cube root of a number is the same as raising it to the power of $\frac{1}{3}$. - The square root of a number is the same as raising it to the power of $\frac{1}{2}$. - When raising a power to another power, multiply the exponents: $$(a^m)^n = a^{m \cdot n}$$. 3. **Step-by-step simplification:** $$\sqrt[3]{8x^7} = (8x^7)^{\frac{1}{3}}$$ 4. If the problem involves taking the square root of the cube root, then: $$\sqrt{\sqrt[3]{8x^7}} = \left((8x^7)^{\frac{1}{3}}\right)^{\frac{1}{2}}$$ 5. Multiply the exponents: $$= (8x^7)^{\frac{1}{3} \cdot \frac{1}{2}} = (8x^7)^{\frac{1}{6}}$$ 6. Apply the exponent to each factor: $$= 8^{\frac{1}{6}} \cdot x^{7 \cdot \frac{1}{6}} = 8^{\frac{1}{6}} x^{\frac{7}{6}}$$ 7. Simplify $8^{\frac{1}{6}}$: Since $8 = 2^3$, $$8^{\frac{1}{6}} = (2^3)^{\frac{1}{6}} = 2^{\frac{3}{6}} = 2^{\frac{1}{2}} = \sqrt{2}$$ 8. **Final simplified expression:** $$\sqrt{2} \cdot x^{\frac{7}{6}}$$ This is the simplified form of the original expression.