1. **Stating the problem:** Simplify the expression $$\sqrt{2} \cdot \sqrt[3]{3}$$. 2. **Recall the rules:** - The square root of a number $x$ is written as $\sqrt{x} = x^{\frac{1}{2}}$. - The cube root of a number $x$ is written as $\sqrt[3]{x} = x^{\frac{1}{3}}$. - When multiplying expressions with the same base, add the exponents. 3. **Rewrite the expression using exponents:** $$\sqrt{2} \cdot \sqrt[3]{3} = 2^{\frac{1}{2}} \cdot 3^{\frac{1}{3}}$$ 4. **Since the bases are different (2 and 3), we cannot combine the terms further by multiplication.** 5. **Final simplified form:** $$2^{\frac{1}{2}} \cdot 3^{\frac{1}{3}}$$ This is the simplest exact form of the expression. **Answer:** $$\sqrt{2} \cdot \sqrt[3]{3} = 2^{\frac{1}{2}} \cdot 3^{\frac{1}{3}}$$