1. Problem: Simplify and analyze the function $f(x) = 6x^3 - 9x + 4$. Step 1: Identify the terms and powers of $x$. Step 2: The function is a cubic polynomial with terms $6x^3$, $-9x$, and constant $4$. Step 3: No further simplification needed. 2. Problem: Simplify $y = 2t^4 - 10t^2 + 13t$. Step 1: Recognize the polynomial terms. Step 2: The function is a quartic polynomial with terms $2t^4$, $-10t^2$, and $13t$. 3. Problem: Simplify $g(z) = 4z^7 - 3z^{-7} + 9z$. Step 1: Note the negative exponent term $-3z^{-7} = -\frac{3}{z^7}$. Step 2: The function combines positive and negative powers of $z$. 4. Problem: Simplify $h(y) = y^{-4} - 9y^{-3} + 8y^{-2} + 12$. Step 1: Rewrite negative exponents as fractions: $y^{-4} = \frac{1}{y^4}$, etc. 5. Problem: Simplify $y = \sqrt{x} + 8 \sqrt[3]{x} - 2 \sqrt[4]{x}$. Step 1: Express roots as fractional exponents: $\sqrt{x} = x^{1/2}$, $\sqrt[3]{x} = x^{1/3}$, $\sqrt[4]{x} = x^{1/4}$. 6. Problem: Simplify $f(x) = 10 \sqrt[5]{x^3} - \sqrt{x^7} + 6 \sqrt[3]{x^8} - 3$. Step 1: Express roots as fractional powers: $\sqrt[5]{x^3} = x^{3/5}$, $\sqrt{x^7} = x^{7/2}$, $\sqrt[3]{x^8} = x^{8/3}$. 7. Problem: Simplify $f(t) = \frac{4}{t} - \frac{1}{6t^3} + \frac{8}{t^5}$. Step 1: Express as powers: $\frac{4}{t} = 4t^{-1}$, etc. 8. Problem: Simplify $R(z) = \frac{6}{\sqrt{z^3}} + \frac{1}{8z^4} - \frac{1}{3z^{10}}$. Step 1: Rewrite $\sqrt{z^3} = z^{3/2}$. 9. Problem: Simplify $z = x(3x^2 - 9)$. Step 1: Distribute: $z = 3x^3 - 9x$. 10. Problem: Simplify $g(y) = (y - 4)(2y + y^2)$. Step 1: Expand: $g(y) = y(2y + y^2) - 4(2y + y^2) = 2y^2 + y^3 - 8y - 4y^2 = y^3 - 2y^2 - 8y$. 11. Problem: Simplify $h(x) = \frac{4x^3 - 7x + 8}{x}$. Step 1: Divide each term by $x$: $h(x) = 4x^2 - 7 + \frac{8}{x}$. 12. Problem: Simplify $f(y) = \frac{y^5 - 5y^3 + 2y}{y^3}$. Step 1: Divide each term: $f(y) = y^{5-3} - 5y^{3-3} + 2y^{1-3} = y^2 - 5 + 2y^{-2}$. 13. Problem: Find where $f(x) = x^3 + 9x^2 - 48x + 2$ is not changing. Step 1: Find derivative: $f'(x) = 3x^2 + 18x - 48$. Step 2: Set $f'(x) = 0$: $3x^2 + 18x - 48 = 0$. Step 3: Divide by 3: $x^2 + 6x - 16 = 0$. Step 4: Solve quadratic: $x = \frac{-6 \pm \sqrt{36 + 64}}{2} = \frac{-6 \pm 10}{2}$. Step 5: Solutions: $x = 2$ or $x = -8$. 14. Problem: Find where $y = 2z^4 - z^3 - 3z^2$ is not changing. Step 1: Derivative: $y' = 8z^3 - 3z^2 - 6z$. Step 2: Set $y' = 0$: $8z^3 - 3z^2 - 6z = 0$. Step 3: Factor out $z$: $z(8z^2 - 3z - 6) = 0$. Step 4: Solutions: $z=0$ or solve $8z^2 - 3z - 6=0$. Step 5: Quadratic formula: $z = \frac{3 \pm \sqrt{9 + 192}}{16} = \frac{3 \pm 15}{16}$. Step 6: Solutions: $z = \frac{18}{16} = \frac{9}{8}$ or $z = \frac{-12}{16} = -\frac{3}{4}$. 15. Problem: Find tangent line to $g(x) = \frac{16}{x} - 4\sqrt{x}$ at $x=4$. Step 1: Rewrite $g(x) = 16x^{-1} - 4x^{1/2}$. Step 2: Derivative: $g'(x) = -16x^{-2} - 2x^{-1/2}$. Step 3: Evaluate $g'(4) = -16/16 - 2/2 = -1 -1 = -2$. Step 4: Evaluate $g(4) = 16/4 - 4*2 = 4 - 8 = -4$. Step 5: Equation of tangent line: $y - (-4) = -2(x - 4)$ or $y = -2x + 8 - 4 = -2x + 4$. 16. Problem: Find tangent line to $f(x) = 7x^4 + 8x^{-6} + 2x$ at $x = -1$. Step 1: Derivative: $f'(x) = 28x^3 - 48x^{-7} + 2$. Step 2: Evaluate $f'(-1) = 28(-1)^3 - 48(-1)^{-7} + 2 = -28 - 48 + 2 = -74$. Step 3: Evaluate $f(-1) = 7(-1)^4 + 8(-1)^{-6} + 2(-1) = 7 + 8 - 2 = 13$. Step 4: Tangent line: $y - 13 = -74(x + 1)$ or $y = -74x - 74 + 13 = -74x - 61$. Final answers: 1. $f(x) = 6x^3 - 9x + 4$ 2. $y = 2t^4 - 10t^2 + 13t$ 3. $g(z) = 4z^7 - \frac{3}{z^7} + 9z$ 4. $h(y) = \frac{1}{y^4} - \frac{9}{y^3} + \frac{8}{y^2} + 12$ 5. $y = x^{1/2} + 8x^{1/3} - 2x^{1/4}$ 6. $f(x) = 10x^{3/5} - x^{7/2} + 6x^{8/3} - 3$ 7. $f(t) = 4t^{-1} - \frac{1}{6}t^{-3} + 8t^{-5}$ 8. $R(z) = 6z^{-3/2} + \frac{1}{8}z^{-4} - \frac{1}{3}z^{-10}$ 9. $z = 3x^3 - 9x$ 10. $g(y) = y^3 - 2y^2 - 8y$ 11. $h(x) = 4x^2 - 7 + \frac{8}{x}$ 12. $f(y) = y^2 - 5 + 2y^{-2}$ 13. Not changing at $x=2$ and $x=-8$ 14. Not changing at $z=0$, $z=\frac{9}{8}$, and $z=-\frac{3}{4}$ 15. Tangent line: $y = -2x + 4$ 16. Tangent line: $y = -74x - 61$