1. Problem: Simplify each expression for parts a, b, c, d, e, f and explain how to do g. 2. Formula and rules: Combine like terms by adding or subtracting coefficients. Multiply powers with the same base by adding exponents. Divide powers with the same base by subtracting exponents. Cancel common factors when dividing. 3. a) Simplify $2k + 8 - k^2 - 4k$: Combine like terms $2k - 4k = -2k$: $$2k + 8 - k^2 - 4k = -k^2 - 2k + 8$$ 4. b) Simplify $ay^2 \times \frac{1}{2} y^2$: Multiply coefficients: $a \times \frac{1}{2} = \frac{a}{2}$ Multiply powers: $y^2 \times y^2 = y^{2+2} = y^4$ Result: $$\frac{a}{2} y^4$$ 5. c) Simplify $-5t \div (-20t)$: Write as fraction: $$\frac{-5t}{-20t}$$ Cancel $t$: $$\frac{-5\cancel{t}}{-20\cancel{t}} = \frac{-5}{-20}$$ Simplify fraction: $$\frac{-5}{-20} = \frac{\cancel{-5}}{\cancel{-20}} = \frac{1}{4}$$ 6. d) Simplify $\frac{-32f^3 g^2}{4f^2 g}$: Divide coefficients: $$\frac{-32}{4} = -8$$ Divide powers: $$f^{3-2} = f^1 = f$$ $$g^{2-1} = g$$ Result: $$-8fg$$ 7. e) Simplify $-3c^2 + 7c - 7c^2 + 6c$: Combine like terms: $$-3c^2 - 7c^2 = -10c^2$$ $$7c + 6c = 13c$$ Result: $$-10c^2 + 13c$$ 8. f) Simplify $4z + z + 8 - 4z^2$: Combine like terms: $$4z + z = 5z$$ Result: $$-4z^2 + 5z + 8$$ 9. How to do g) Simplify $-3a \times (-2a^2)$: Multiply coefficients: $$-3 \times -2 = 6$$ Multiply powers: $$a \times a^2 = a^{1+2} = a^3$$ Result: $$6a^3$$ Final answers: a) $-k^2 - 2k + 8$ b) $\frac{a}{2} y^4$ c) $\frac{1}{4}$ d) $-8fg$ e) $-10c^2 + 13c$ f) $-4z^2 + 5z + 8$ g) $6a^3$ (method shown)