Radical Simplification Ec5803

1. We are asked to write each expression in radical form and simplify. 2. Recall the rule: $a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$. 3. Problem 5: Simplify $16^{3/4}$. - Write in radical form: $16^{3/4} = \left(\sqrt[4]{16}\right)^3$. - Since $16 = 2^4$, $\sqrt[4]{16} = 2$. - So, $16^{3/4} = 2^3 = 8$. 4. Problem 6: Simplify $7^{1/3}$. - Write in radical form: $7^{1/3} = \sqrt[3]{7}$. - This is already simplified as $7$ is not a perfect cube. 5. Problem 7: Simplify $-1000^{2/3}$. - Note the negative sign is outside the power: $-(1000^{2/3})$. - Write in radical form: $1000^{2/3} = \left(\sqrt[3]{1000}\right)^2$. - Since $1000 = 10^3$, $\sqrt[3]{1000} = 10$. - So, $1000^{2/3} = 10^2 = 100$. - Therefore, $-1000^{2/3} = -100$. 6. Problem 8: Simplify $4^{5/2}$. - Write in radical form: $4^{5/2} = \left(\sqrt{4}\right)^5$. - Since $\sqrt{4} = 2$, $4^{5/2} = 2^5 = 32$. **Final answers:** - $16^{3/4} = 8$ - $7^{1/3} = \sqrt[3]{7}$ - $-1000^{2/3} = -100$ - $4^{5/2} = 32$