1. Simplify each expression step-by-step. **a. Simplify** $\sqrt{36x^4}$ 1. The square root of a product is the product of the square roots: $\sqrt{36x^4} = \sqrt{36} \cdot \sqrt{x^4}$. 2. Calculate $\sqrt{36} = 6$ because $6^2 = 36$. 3. For $\sqrt{x^4}$, use the rule $\sqrt{x^{2n}} = x^n$, so $\sqrt{x^4} = x^{4/2} = x^2$. 4. Therefore, $\sqrt{36x^4} = 6x^2$. **b. Simplify** $\sqrt[3]{-8x^3}$ 1. The cube root of a product is the product of the cube roots: $\sqrt[3]{-8x^3} = \sqrt[3]{-8} \cdot \sqrt[3]{x^3}$. 2. Calculate $\sqrt[3]{-8} = -2$ because $(-2)^3 = -8$. 3. For $\sqrt[3]{x^3}$, use the rule $\sqrt[3]{x^3} = x^{3/3} = x$. 4. Therefore, $\sqrt[3]{-8x^3} = -2x$. **c. Simplify** $27^{2/3} + 16^{3/4}$ 1. Recall $a^{m/n} = \sqrt[n]{a^m}$. 2. Simplify $27^{2/3}$: - $27 = 3^3$, so $27^{2/3} = (3^3)^{2/3} = 3^{3 \cdot \frac{2}{3}} = 3^2 = 9$. 3. Simplify $16^{3/4}$: - $16 = 2^4$, so $16^{3/4} = (2^4)^{3/4} = 2^{4 \cdot \frac{3}{4}} = 2^3 = 8$. 4. Add the results: $9 + 8 = 17$. **d. Simplify** $16^{-3/4}$ 1. Use the negative exponent rule: $a^{-m} = \frac{1}{a^m}$. 2. So, $16^{-3/4} = \frac{1}{16^{3/4}}$. 3. Simplify $16^{3/4}$ as above: $16^{3/4} = 8$. 4. Therefore, $16^{-3/4} = \frac{1}{8}$. **e. Simplify** $\sqrt[5]{x^{10} y^{15}}$ 1. Use the rule $\sqrt[n]{a^m} = a^{m/n}$. 2. So, $\sqrt[5]{x^{10} y^{15}} = x^{10/5} y^{15/5} = x^2 y^3$. **Final answers:** - a. $6x^2$ - b. $-2x$ - c. $17$ - d. $\frac{1}{8}$ - e. $x^2 y^3$