1. **State the problem:** Evaluate the expression $$\sqrt[3]{\sqrt{256} + \sqrt{81}} + \sqrt[3]{512 + \sqrt[3]{1}}$$ and simplify the answer. 2. **Recall important rules:** - The square root $\sqrt{x}$ is the number that when squared gives $x$. - The cube root $\sqrt[3]{x}$ is the number that when cubed gives $x$. - Simplify inside the roots first before applying cube roots. 3. **Simplify inside the first cube root:** - $\sqrt{256} = 16$ because $16^2 = 256$. - $\sqrt{81} = 9$ because $9^2 = 81$. - So, inside the first cube root: $16 + 9 = 25$. 4. **Simplify inside the second cube root:** - $\sqrt[3]{1} = 1$ because $1^3 = 1$. - So, inside the second cube root: $512 + 1 = 513$. 5. **Rewrite the expression:** $$\sqrt[3]{25} + \sqrt[3]{513}$$ 6. **Evaluate cube roots if possible:** - $\sqrt[3]{25}$ is not a perfect cube, so leave as is. - $\sqrt[3]{513}$ is not a perfect cube either. 7. **Final simplified expression:** $$\sqrt[3]{25} + \sqrt[3]{513}$$ This is the simplest exact form. **Answer:** $$\sqrt[3]{25} + \sqrt[3]{513}$$