1. **Simplify** $16^{-\frac{3}{4}}$. Recall the rule: $a^{-m} = \frac{1}{a^m}$. Since $16 = 2^4$, rewrite: $$16^{-\frac{3}{4}} = (2^4)^{-\frac{3}{4}} = 2^{4 \times -\frac{3}{4}} = 2^{-3} = \frac{1}{2^3} = \frac{1}{8}.$$ 2. **Simplify** $\sqrt[5]{x^{10} y^{15}}$. Use the rule: $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. Rewrite: $$\sqrt[5]{x^{10} y^{15}} = x^{\frac{10}{5}} y^{\frac{15}{5}} = x^2 y^3.$$ 3. **Simplify and express in radical form:** $2\sqrt{9} - 5\sqrt{3} - \sqrt{75}$. Calculate each term: $\sqrt{9} = 3$, so $2\sqrt{9} = 2 \times 3 = 6$. Simplify $\sqrt{75}$: $$\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}.$$ Rewrite expression: $$6 - 5\sqrt{3} - 5\sqrt{3} = 6 - 10\sqrt{3}.$$ 4. **Simplify and express in radical form:** $\sqrt{125} + 3\sqrt{50} - 2\sqrt{5}$. Simplify radicals: $$\sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5},$$ $$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}.$$ Rewrite expression: $$5\sqrt{5} + 3 \times 5\sqrt{2} - 2\sqrt{5} = 5\sqrt{5} + 15\sqrt{2} - 2\sqrt{5}.$$ Combine like terms: $$5\sqrt{5} - 2\sqrt{5} = 3\sqrt{5}.$$ Final expression: $$3\sqrt{5} + 15\sqrt{2}.$$ 5. **Simplify and express in radical form:** $4\sqrt{63} - 3\sqrt{7} + \sqrt{28}$. Simplify radicals: $$\sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7},$$ $$\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}.$$ Rewrite expression: $$4 \times 3\sqrt{7} - 3\sqrt{7} + 2\sqrt{7} = 12\sqrt{7} - 3\sqrt{7} + 2\sqrt{7}.$$ Combine like terms: $$12\sqrt{7} - 3\sqrt{7} + 2\sqrt{7} = (12 - 3 + 2)\sqrt{7} = 11\sqrt{7}.$$ **Final answers:** - d. $16^{-\frac{3}{4}} = \frac{1}{8}$ - e. $\sqrt[5]{x^{10} y^{15}} = x^2 y^3$ - a. $2\sqrt{9} - 5\sqrt{3} - \sqrt{75} = 6 - 10\sqrt{3}$ - b. $\sqrt{125} + 3\sqrt{50} - 2\sqrt{5} = 3\sqrt{5} + 15\sqrt{2}$ - c. $4\sqrt{63} - 3\sqrt{7} + \sqrt{28} = 11\sqrt{7}$