1. **Express as a single power and then evaluate:** **a)** $\frac{(-5)^6}{(-5)^3} = (-5)^{6-3} = (-5)^3$ Calculate $(-5)^3 = -125$ **b)** $(2y^4)^4 = 2^4 \cdot (y^4)^4 = 16y^{16}$ **c)** Simplify $-\frac{(4x^2y^2)^2}{(-2x^3y^2)^2p}$: Calculate numerator: $(4x^2y^2)^2 = 4^2 x^{4} y^{4} = 16x^{4}y^{4}$ Calculate denominator: $(-2x^3y^2)^2 p = (-2)^2 x^{6} y^{4} p = 4x^{6} y^{4} p$ Divide numerator by denominator: $$-\frac{16x^{4}y^{4}}{4x^{6}y^{4}p} = -\frac{\cancel{16}x^{4}y^{4}}{\cancel{4}x^{6}y^{4}p} = -\frac{4}{x^{2}p}$$ Without values for $x$ and $p$, this is the simplified form. 2. **Evaluate:** $(3a^3)^2 (3^3 a^6 b^2)^2$ Calculate each term: $(3a^3)^2 = 3^2 a^{6} = 9a^{6}$ $(3^3 a^6 b^2)^2 = 3^{6} a^{12} b^{4}$ Multiply: $9a^{6} \cdot 3^{6} a^{12} b^{4} = 9 \cdot 729 \cdot a^{18} b^{4} = 6561 a^{18} b^{4}$ 3. **Write with a common base and simplify:** $\frac{81^2}{3^6}$ Since $81 = 3^4$, rewrite numerator: $\frac{(3^4)^2}{3^6} = \frac{3^{8}}{3^{6}} = 3^{8-6} = 3^{2} = 9$ 4. **Simplify by distributive property:** **a)** $2b(3b - 5) + 2 = 2b \cdot 3b - 2b \cdot 5 + 2 = 6b^{2} - 10b + 2$ **b)** $\frac{28mn + 14n}{7n} = \frac{28mn}{7n} + \frac{14n}{7n} = 4m + 2$ 5. **Create equation and solve for $x$: given angles $(3x - 31)^{\circ}$ and $(x + 59)^{\circ}$ are supplementary:** Sum of angles = $180^{\circ}$ $$ (3x - 31) + (x + 59) = 180 $$ Simplify: $$ 3x - 31 + x + 59 = 180 $$ $$ 4x + 28 = 180 $$ $$ 4x = 180 - 28 = 152 $$ $$ x = \frac{152}{4} = 38 $$ **Final answers:** a) $-125$ b) $16y^{16}$ c) $-\frac{4}{x^{2}p}$ (simplified form) 2) $6561 a^{18} b^{4}$ 3) $9$ 4a) $6b^{2} - 10b + 2$ 4b) $4m + 2$ 5) $x = 38$