1. **Express each radical as a mixed radical in simplest form:** 1.a. Simplify $\sqrt{56}$: - Factor 56 into prime factors: $56 = 4 \times 14 = 2^2 \times 14$ - Use the property $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$: $$\sqrt{56} = \sqrt{4 \times 14} = \sqrt{4} \times \sqrt{14} = 2\sqrt{14}$$ 1.b. Simplify $3\sqrt{75}$: - Factor 75: $75 = 25 \times 3 = 5^2 \times 3$ - Simplify inside the cube root: $$3\sqrt{75} = 3\sqrt{25 \times 3} = 3 \times 3\sqrt{25} \times 3\sqrt{3}$$ - But since $3\sqrt{a}$ means cube root, rewrite as $\sqrt[3]{75}$: - Actually, the problem likely means cube root of 75, so: $$\sqrt[3]{75} = \sqrt[3]{25 \times 3}$$ - No perfect cube factors in 25 or 3, so leave as is or factor 75 as $3 \times 25$. - Since 25 is not a perfect cube, the simplest form is $\sqrt[3]{75}$. 1.c. Simplify $3\sqrt{8m^4}$: - Factor inside the cube root: $8m^4 = 8 \times m^3 \times m = 2^3 \times m^3 \times m$ - Use property $\sqrt[3]{a^3} = a$: $$3\sqrt{8m^4} = 3\sqrt{2^3 m^3 m} = 3\sqrt{2^3} \times 3\sqrt{m^3} \times 3\sqrt{m} = 2m 3\sqrt{m}$$ 1.d. Simplify $\sqrt[3]{24q^5}$: - Factor inside the cube root: $24q^5 = 8 \times 3 \times q^3 \times q^2$ - Use property $\sqrt[3]{a^3} = a$: $$\sqrt[3]{24q^5} = \sqrt[3]{8 \times 3 \times q^3 \times q^2} = \sqrt[3]{8} \times \sqrt[3]{3} \times \sqrt[3]{q^3} \times \sqrt[3]{q^2} = 2q \sqrt[3]{3q^2}$$ 2. **Simplify each expression:** 2.a. Simplify $3\sqrt{75} - \sqrt{27}$: - Simplify each radical: $$3\sqrt{75} = 3 \times 5\sqrt[3]{3} = 15\sqrt[3]{3}$$ $$\sqrt{27} = 3\sqrt{3}$$ - Since $\sqrt[3]{3}$ and $\sqrt{3}$ are different radicals, leave as is: $$15\sqrt[3]{3} - 3\sqrt{3}$$ 2.b. Simplify $2\sqrt{18} + 9\sqrt{7} - \sqrt{63}$: - Simplify radicals: $$2\sqrt{18} = 2 \times 3\sqrt{2} = 6\sqrt{2}$$ $$9\sqrt{7} = 9\sqrt{7}$$ $$\sqrt{63} = 3\sqrt{7}$$ - Combine like terms: $$6\sqrt{2} + 9\sqrt{7} - 3\sqrt{7} = 6\sqrt{2} + 6\sqrt{7}$$ 2.c. Simplify $3\sqrt{2x} + 3\sqrt{8x} - \sqrt{x}$: - Simplify $3\sqrt{8x}$: $$3\sqrt{8x} = 3 \times 2\sqrt{2x} = 6\sqrt{2x}$$ - Now expression is: $$3\sqrt{2x} + 6\sqrt{2x} - \sqrt{x} = 9\sqrt{2x} - \sqrt{x}$$ 2.d. Simplify $5\sqrt{3} \times \sqrt{6}$: - Multiply under one radical: $$5\sqrt{3} \times \sqrt{6} = 5\sqrt{3 \times 6} = 5\sqrt{18}$$ - Simplify $\sqrt{18} = 3\sqrt{2}$: $$5 \times 3\sqrt{2} = 15\sqrt{2}$$ 2.e. Simplify $-2\sqrt[3]{11}(4\sqrt[3]{2} - 3\sqrt{3})$: - Distribute: $$-2\sqrt[3]{11} \times 4\sqrt[3]{2} = -8\sqrt[3]{22}$$ $$-2\sqrt[3]{11} \times (-3\sqrt{3}) = +6\sqrt[3]{11} \sqrt{3}$$ - Since $\sqrt[3]{11} \sqrt{3}$ cannot be combined simply, leave as is: $$-8\sqrt[3]{22} + 6\sqrt[3]{33}$$ 2.f. Simplify $(4\sqrt{2} + 3)(\sqrt{7} - 5\sqrt{14})$: - Use distributive property: $$4\sqrt{2} \times \sqrt{7} = 4\sqrt{14}$$ $$4\sqrt{2} \times (-5\sqrt{14}) = -20\sqrt{28} = -20 \times 2\sqrt{7} = -40\sqrt{7}$$ $$3 \times \sqrt{7} = 3\sqrt{7}$$ $$3 \times (-5\sqrt{14}) = -15\sqrt{14}$$ - Combine like terms: $$(4\sqrt{14} - 15\sqrt{14}) + (-40\sqrt{7} + 3\sqrt{7}) = -11\sqrt{14} - 37\sqrt{7}$$