1. Problem 1: Simplify the expression $\frac{1}{3}x - (8\sqrt{x} + \sqrt{18x} + \sqrt{2x})$. - Recall that $\sqrt{a} = a^{1/2}$ and simplify each term inside the parentheses. - $\sqrt{18x} = \sqrt{9 \cdot 2x} = 3\sqrt{2x}$. - So the expression becomes $\frac{1}{3}x - (8x^{1/2} + 3\sqrt{2x} + \sqrt{2x}) = \frac{1}{3}x - (8x^{1/2} + 4\sqrt{2x})$. 2. Problem 2: Simplify $\frac{1}{3^2}(16 - 4^{3/2})$. - $3^2 = 9$. - $4^{3/2} = (4^{1/2})^3 = 2^3 = 8$. - So the expression is $\frac{1}{9}(16 - 8) = \frac{8}{9}$. 3. Problem 3: Simplify $(3\sqrt{x} - y) - \sqrt{2}x^4 + 2x^3y$. - This is already simplified; just rewrite as $3x^{1/2} - y - \sqrt{2}x^4 + 2x^3y$. 4. Problem 4a: Simplify $\sqrt{3} \cdot \sqrt[3]{3} \cdot \sqrt[4]{3}$. - Express all as powers: $3^{1/2} \cdot 3^{1/3} \cdot 3^{1/4} = 3^{1/2 + 1/3 + 1/4}$. - Find common denominator 12: $\frac{6}{12} + \frac{4}{12} + \frac{3}{12} = \frac{13}{12}$. - Final: $3^{13/12}$. 5. Problem 4c: Simplify $(\sqrt[6]{2} + 2^{3/4} - \sqrt{8}) \cdot \sqrt[6]{32}$. - $\sqrt[6]{2} = 2^{1/6}$. - $2^{3/4}$ stays as is. - $\sqrt{8} = \sqrt{4 \cdot 2} = 2 \sqrt{2} = 2 \cdot 2^{1/2} = 2^{3/2}$. - $\sqrt[6]{32} = 32^{1/6} = (2^5)^{1/6} = 2^{5/6}$. - So expression inside parentheses: $2^{1/6} + 2^{3/4} - 2^{3/2}$. - Multiply entire sum by $2^{5/6}$. - Result: $2^{1/6 + 5/6} + 2^{3/4 + 5/6} - 2^{3/2 + 5/6} = 2^{1} + 2^{19/12} - 2^{13/6}$. 6. Problem 4b (first): Simplify $a^2 b^6 16 a^5 b \cdot \frac{1}{a} \sqrt{2ab} \cdot \sqrt[3]{4ab^2}$. - Combine powers: $a^{2+5} b^{6+1} 16 = 16 a^7 b^7$. - $\sqrt{2ab} = (2ab)^{1/2} = 2^{1/2} a^{1/2} b^{1/2}$. - $\sqrt[3]{4ab^2} = (4ab^2)^{1/3} = 4^{1/3} a^{1/3} b^{2/3}$. - Multiply all: $16 a^7 b^7 \cdot \frac{1}{a} \cdot 2^{1/2} a^{1/2} b^{1/2} \cdot 4^{1/3} a^{1/3} b^{2/3}$. - Simplify $a$ powers: $a^{7 - 1 + 1/2 + 1/3} = a^{6 + 1/2 + 1/3} = a^{6 + 5/6} = a^{41/6}$. - Simplify $b$ powers: $b^{7 + 1/2 + 2/3} = b^{7 + 3/6 + 4/6} = b^{7 + 7/6} = b^{49/6}$. - Constants: $16 \cdot 2^{1/2} \cdot 4^{1/3} = 16 \cdot 2^{1/2} \cdot (2^2)^{1/3} = 16 \cdot 2^{1/2} \cdot 2^{2/3} = 16 \cdot 2^{7/6} = 2^4 \cdot 2^{7/6} = 2^{4 + 7/6} = 2^{31/6}$. - Final: $2^{31/6} a^{41/6} b^{49/6}$. 7. Problem 4b (second): Simplify $(4x^3 x^2 - 5 y^3 \sqrt{x y} + x y^3 \sqrt{y^2}) x y^3 \sqrt{x y}$. - Simplify inside parentheses: - $4x^3 x^2 = 4x^{5}$. - $\sqrt{x y} = (x y)^{1/2} = x^{1/2} y^{1/2}$. - $-5 y^3 \sqrt{x y} = -5 y^3 x^{1/2} y^{1/2} = -5 x^{1/2} y^{3 + 1/2} = -5 x^{1/2} y^{7/2}$. - $\sqrt{y^2} = y$. - $x y^3 \sqrt{y^2} = x y^3 y = x y^{4}$. - So parentheses: $4 x^{5} - 5 x^{1/2} y^{7/2} + x y^{4}$. - Multiply by $x y^{3} \sqrt{x y} = x y^{3} x^{1/2} y^{1/2} = x^{3/2} y^{3 + 1/2} = x^{3/2} y^{7/2}$. - Multiply each term: - $4 x^{5} \cdot x^{3/2} y^{7/2} = 4 x^{5 + 3/2} y^{7/2} = 4 x^{13/2} y^{7/2}$. - $-5 x^{1/2} y^{7/2} \cdot x^{3/2} y^{7/2} = -5 x^{1/2 + 3/2} y^{7/2 + 7/2} = -5 x^{2} y^{7}$. - $x y^{4} \cdot x^{3/2} y^{7/2} = x^{1 + 3/2} y^{4 + 7/2} = x^{5/2} y^{15/2}$. Final simplified expression: $4 x^{13/2} y^{7/2} - 5 x^{2} y^{7} + x^{5/2} y^{15/2}$. --- Slug: "radical expressions" Subject: "algebra" Desmos: {"latex":"y=0","features":{"intercepts":true,"extrema":true}} q_count: 7