1. **State the problem:** Simplify the expression $$\frac{\sqrt[3]{2\sqrt{8}}}{\sqrt[3]{\sqrt{2\sqrt{2}}}}$$. 2. **Recall the rules:** - The cube root of a product is the product of the cube roots: $$\sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b}$$. - The square root of a product is the product of the square roots: $$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$$. - Powers can be rewritten as fractional exponents: $$\sqrt[n]{x} = x^{\frac{1}{n}}$$. 3. **Rewrite the expression using fractional exponents:** - $$\sqrt[3]{2\sqrt{8}} = (2 \cdot 8^{\frac{1}{2}})^{\frac{1}{3}} = (2 \cdot 8^{0.5})^{\frac{1}{3}}$$ - $$\sqrt[3]{\sqrt{2\sqrt{2}}} = \left((2 \cdot 2^{\frac{1}{2}})^{\frac{1}{2}}\right)^{\frac{1}{3}} = (2 \cdot 2^{0.5})^{\frac{1}{2} \cdot \frac{1}{3}} = (2 \cdot 2^{0.5})^{\frac{1}{6}}$$ 4. **Simplify inside the roots:** - $$8^{0.5} = (2^3)^{0.5} = 2^{3 \times 0.5} = 2^{1.5}$$ - So numerator inside: $$2 \cdot 8^{0.5} = 2 \cdot 2^{1.5} = 2^{1} \cdot 2^{1.5} = 2^{2.5}$$ - Denominator inside: $$2 \cdot 2^{0.5} = 2^{1} \cdot 2^{0.5} = 2^{1.5}$$ 5. **Apply the outer exponents:** - Numerator: $$\left(2^{2.5}\right)^{\frac{1}{3}} = 2^{\frac{2.5}{3}} = 2^{\frac{5}{6}}$$ - Denominator: $$\left(2^{1.5}\right)^{\frac{1}{6}} = 2^{\frac{1.5}{6}} = 2^{\frac{1}{4}}$$ 6. **Divide the powers of 2:** $$\frac{2^{\frac{5}{6}}}{2^{\frac{1}{4}}} = 2^{\frac{5}{6} - \frac{1}{4}} = 2^{\frac{10}{12} - \frac{3}{12}} = 2^{\frac{7}{12}}$$ **Final answer:** $$2^{\frac{7}{12}}$$