1. Express each radical as a mixed radical in simplest form: 1.a. Simplify $\sqrt{56}$: - Factor 56 into $7 \times 8 = 7 \times 2^3$ - Use $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ - $\sqrt{56} = \sqrt{7 \times 2^3} = \sqrt{7} \times \sqrt{2^3} = \sqrt{7} \times 2\sqrt{2} = 2\sqrt{14}$ 1.b. Simplify $\sqrt[3]{75}$: - Factor 75 into $3 \times 25 = 3 \times 5^2$ - $\sqrt[3]{75} = \sqrt[3]{3 \times 5^2}$ - No perfect cube factors, so leave as $\sqrt[3]{75}$ or write as $\sqrt[3]{3 \times 5^2}$ 1.c. Simplify $\sqrt[3]{8m^4}$: - Factor $8 = 2^3$ - $m^4 = m^3 \times m$ - $\sqrt[3]{8m^4} = \sqrt[3]{2^3 m^3 m} = 2m \sqrt[3]{m}$ 1.d. Simplify $\sqrt[3]{24q^5}$: - Factor $24 = 2^3 \times 3$ - $q^5 = q^3 \times q^2$ - $\sqrt[3]{24q^5} = \sqrt[3]{2^3 \times 3 \times q^3 \times q^2} = 2q \sqrt[3]{3q^2}$ 2. Simplify each expression: 2.a. $3\sqrt[3]{75} - \sqrt{27}$ - From 1.b, $\sqrt[3]{75}$ remains as is - Simplify $\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$ - Expression: $3\sqrt[3]{75} - 3\sqrt{3}$ (cannot combine cube root and square root terms) 2.b. $2\sqrt{18} + 9\sqrt{7} - \sqrt{63}$ - Simplify $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$ - Simplify $\sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7}$ - Expression: $2 \times 3\sqrt{2} + 9\sqrt{7} - 3\sqrt{7} = 6\sqrt{2} + 6\sqrt{7}$ 2.c. $3\sqrt{2x} + 3\sqrt{8x} - \sqrt{x}$ - Simplify $\sqrt{8x} = \sqrt{4 \times 2x} = 2\sqrt{2x}$ - Expression: $3\sqrt{2x} + 3 \times 2\sqrt{2x} - \sqrt{x} = 3\sqrt{2x} + 6\sqrt{2x} - \sqrt{x} = 9\sqrt{2x} - \sqrt{x}$ 2.d. $5\sqrt{3} \sqrt{6}$ - Multiply under one radical: $\sqrt{3} \times \sqrt{6} = \sqrt{18} = 3\sqrt{2}$ - Expression: $5 \times 3\sqrt{2} = 15\sqrt{2}$ 2.e. $-2\sqrt[3]{11}(4\sqrt[3]{2} - 3\sqrt{3})$ - Distribute: $-2\sqrt[3]{11} \times 4\sqrt[3]{2} = -8 \sqrt[3]{11 \times 2} = -8 \sqrt[3]{22}$ $-2\sqrt[3]{11} \times (-3\sqrt{3}) = +6 \sqrt[3]{11} \sqrt{3}$ (cannot combine cube root and square root) - Expression: $-8 \sqrt[3]{22} + 6 \sqrt[3]{11} \sqrt{3}$ 2.f. $(4\sqrt{2} + 3)(\sqrt{7} - 5\sqrt{14})$ - Multiply each term: $4\sqrt{2} \times \sqrt{7} = 4\sqrt{14}$ $4\sqrt{2} \times (-5\sqrt{14}) = -20 \sqrt{28} = -20 \sqrt{4 \times 7} = -20 \times 2 \sqrt{7} = -40\sqrt{7}$ $3 \times \sqrt{7} = 3\sqrt{7}$ $3 \times (-5\sqrt{14}) = -15\sqrt{14}$ - Combine like terms: $4\sqrt{14} - 15\sqrt{14} = -11\sqrt{14}$ $-40\sqrt{7} + 3\sqrt{7} = -37\sqrt{7}$ - Final expression: $-11\sqrt{14} - 37\sqrt{7}$