1. **Problem:** Simplify the expression $$\sqrt[4]{\sqrt{2}} \cdot \sqrt[3]{4} \cdot \sqrt[5]{32}$$. 2. **Rewrite each radical as an exponent:** - $$\sqrt[4]{\sqrt{2}} = \left(2^{\frac{1}{2}}\right)^{\frac{1}{4}} = 2^{\frac{1}{2} \cdot \frac{1}{4}} = 2^{\frac{1}{8}}$$ - $$\sqrt[3]{4} = 4^{\frac{1}{3}} = \left(2^2\right)^{\frac{1}{3}} = 2^{\frac{2}{3}}$$ - $$\sqrt[5]{32} = 32^{\frac{1}{5}} = \left(2^5\right)^{\frac{1}{5}} = 2^{1}$$ 3. **Multiply the expressions by adding exponents:** $$2^{\frac{1}{8}} \cdot 2^{\frac{2}{3}} \cdot 2^{1} = 2^{\frac{1}{8} + \frac{2}{3} + 1}$$ 4. **Find common denominator and add exponents:** - Common denominator is 24. - $$\frac{1}{8} = \frac{3}{24}, \quad \frac{2}{3} = \frac{16}{24}, \quad 1 = \frac{24}{24}$$ - Sum: $$\frac{3}{24} + \frac{16}{24} + \frac{24}{24} = \frac{43}{24}$$ 5. **Final simplified form:** $$2^{\frac{43}{24}}$$