1. Problem (a): Simplify $$y = 9 - 15x + 4^3 - \frac{7x^5}{x^4}$$ Step 1: Calculate powers and simplify fractions $$4^3 = 64$$ $$\frac{7x^5}{x^4} = 7x^{5-4} = 7x$$ Step 2: Rewrite expression $$y = 9 - 15x + 64 - 7x$$ Step 3: Combine like terms $$y = (9 + 64) - (15x + 7x) = 73 - 22x$$ --- 2. Problem (b): Evaluate $$f(8)$$ if $$f(t) = -4\sqrt{t^3} - 1$$ Step 1: Substitute $$t=8$$ $$f(8) = -4\sqrt{8^3} - 1$$ Step 2: Calculate inside the square root $$8^3 = 512$$ Step 3: Simplify square root $$\sqrt{512} = \sqrt{256 \times 2} = 16\sqrt{2}$$ Step 4: Calculate $$f(8)$$ $$f(8) = -4 \times 16\sqrt{2} - 1 = -64\sqrt{2} - 1$$ --- 3. Problem (c): Expand $$f(x) = (5x - 2)(3x^2 + 2)$$ Step 1: Use distributive property $$f(x) = 5x \times 3x^2 + 5x \times 2 - 2 \times 3x^2 - 2 \times 2$$ Step 2: Multiply terms $$ = 15x^3 + 10x - 6x^2 - 4$$ Step 3: Write in standard form $$f(x) = 15x^3 - 6x^2 + 10x - 4$$ --- 4. Problem (d): Expand $$y = (\sqrt[3]{x} - 5)(2\sqrt{x} + x^3 + 7)$$ Step 1: Distribute each term $$y = \sqrt[3]{x} \times 2\sqrt{x} + \sqrt[3]{x} \times x^3 + \sqrt[3]{x} \times 7 - 5 \times 2\sqrt{x} - 5 \times x^3 - 5 \times 7$$ Step 2: Simplify each product Recall $$\sqrt[3]{x} = x^{1/3}$$ and $$\sqrt{x} = x^{1/2}$$ $$x^{1/3} \times x^{1/2} = x^{1/3 + 1/2} = x^{5/6}$$ $$x^{1/3} \times x^{3} = x^{1/3 + 3} = x^{10/3}$$ So, $$y = 2x^{5/6} + x^{10/3} + 7x^{1/3} - 10x^{1/2} - 5x^3 - 35$$ --- 5. Problem (e): Simplify $$y = 5 + 2x - 7x^2 (6x^3 - x^2 + 3x + 5)^2$$ Step 1: Identify the complicated part as $$A = (6x^3 - x^2 + 3x + 5)$$ Step 2: The expression is $$y = 5 + 2x - 7x^2 A^2$$ Expanding $$A^2$$ fully is complex; typically, this remains as is unless asked for full expansion. Step 3: Hence, $$y = 5 + 2x - 7x^2 (6x^3 - x^2 + 3x + 5)^2$$ is the simplified expression unless expansion is needed. --- 6. Problem (f): Simplify $$y = \frac{8x^2 - 5}{x^2 - 3}$$ Step 1: There are no common factors between numerator $$8x^2 - 5$$ and denominator $$x^2 - 3$$, so this is the simplified rational expression. --- Final answers: (a) $$y = 73 - 22x$$ (b) $$f(8) = -64\sqrt{2} - 1$$ (c) $$f(x) = 15x^3 - 6x^2 + 10x - 4$$ (d) $$y = 2x^{5/6} + x^{10/3} + 7x^{1/3} - 10x^{1/2} - 5x^3 - 35$$ (e) $$y = 5 + 2x - 7x^2 (6x^3 - x^2 + 3x + 5)^2$$ (f) $$y = \frac{8x^2 - 5}{x^2 - 3}$$