1. **State the problem:** We need to find the values of the square roots and cube roots without using a calculator. 2. **Recall the definitions:** - The square root $\sqrt{a}$ is a number which when multiplied by itself gives $a$. - The cube root $\sqrt[3]{a}$ is a number which when multiplied by itself three times gives $a$. 3. **Calculate each root:** - $\sqrt{121}$ means find a number $x$ such that $x^2 = 121$. - $\sqrt{900}$ means find a number $x$ such that $x^2 = 900$. - $\sqrt[3]{125}$ means find a number $x$ such that $x^3 = 125$. - $\sqrt[3]{729}$ means find a number $x$ such that $x^3 = 729$. 4. **Evaluate $\sqrt{121}$:** Since $11 \times 11 = 121$, we have $$\sqrt{121} = 11$$ 5. **Evaluate $\sqrt{900}$:** Since $30 \times 30 = 900$, we have $$\sqrt{900} = 30$$ 6. **Evaluate $\sqrt[3]{125}$:** Since $5 \times 5 \times 5 = 125$, we have $$\sqrt[3]{125} = 5$$ 7. **Evaluate $\sqrt[3]{729}$:** Since $9 \times 9 \times 9 = 729$, we have $$\sqrt[3]{729} = 9$$ **Final answers:** $$\sqrt{121} = 11, \quad \sqrt{900} = 30, \quad \sqrt[3]{125} = 5, \quad \sqrt[3]{729} = 9$$