1. **Write the statements into algebraic expressions:** (i). Four times the sum of $x$ and three: $$4(x + 3)$$ (ii). Nine times the product of $a$ and $b$, less five times $c$, all divided by seven times $d$: $$\frac{9ab - 5c}{7d}$$ (iii). Sixteen greater than the product of $p$ and $q$: $$pq + 16$$ 2. **Given $a = -2$, $b = 7$, $c = -3$, find the value of:** (i). $$\frac{4b^2 - 2ac}{a + b + c}$$ Step 1: Substitute values: $$\frac{4(7)^2 - 2(-2)(-3)}{-2 + 7 - 3}$$ Step 2: Calculate powers and products: $$\frac{4 \times 49 - 2 \times 6}{2} = \frac{196 - 12}{2}$$ Step 3: Simplify numerator: $$\frac{184}{2}$$ Step 4: Simplify fraction: $$\frac{\cancel{184}}{\cancel{2}} = 92$$ (ii). $$\frac{6b^3 c^2}{5a^4}$$ Step 1: Substitute values: $$\frac{6(7)^3 (-3)^2}{5(-2)^4}$$ Step 2: Calculate powers: $$\frac{6 \times 343 \times 9}{5 \times 16} = \frac{18522}{80}$$ Step 3: Simplify fraction by dividing numerator and denominator by 2: $$\frac{\cancel{18522}/2}{\cancel{80}/2} = \frac{9261}{40}$$ 3. **Simplify the expression:** $$\frac{36x^{-9} y^{7} z^{-2}}{-3xy^{-5/8} \times 2x^{-3} y^{2}}$$ Step 1: Simplify denominator multiplication: $$-3xy^{-5/8} \times 2x^{-3} y^{2} = -6 x^{1 + (-3)} y^{-5/8 + 2} = -6 x^{-2} y^{11/8}$$ Step 2: Write full expression: $$\frac{36 x^{-9} y^{7} z^{-2}}{-6 x^{-2} y^{11/8}}$$ Step 3: Divide coefficients: $$\frac{36}{-6} = -6$$ Step 4: Subtract exponents for $x$ and $y$: $$x^{-9 - (-2)} = x^{-7}$$ $$y^{7 - 11/8} = y^{(56/8 - 11/8)} = y^{45/8}$$ Step 5: Bring down $z^{-2}$: $$z^{-2}$$ Step 6: Combine all: $$-6 x^{-7} y^{45/8} z^{-2}$$ Step 7: Rewrite with positive exponents: $$-6 \frac{y^{45/8}}{x^{7} z^{2}}$$ 4. **Expand and simplify to show:** $$(2x + 3)^2 - (x + 4)(x - 4) \equiv 3x^2 + 12x + 25$$ Step 1: Expand each term: $$(2x + 3)^2 = (2x)^2 + 2 \times 2x \times 3 + 3^2 = 4x^2 + 12x + 9$$ $$(x + 4)(x - 4) = x^2 - 16$$ Step 2: Substitute back: $$4x^2 + 12x + 9 - (x^2 - 16)$$ Step 3: Distribute minus sign: $$4x^2 + 12x + 9 - x^2 + 16$$ Step 4: Combine like terms: $$ (4x^2 - x^2) + 12x + (9 + 16) = 3x^2 + 12x + 25$$ This matches the right side, so the identity is proven.