1. Simplify $2x^5 y \cdot 3x^2 y$. Use the product rule for exponents: $a^m \cdot a^n = a^{m+n}$. $$2x^5 y \cdot 3x^2 y = (2 \cdot 3)(x^{5+2})(y^{1+1}) = 6x^7 y^2$$ 2. Simplify $(4x^4 y)^2 \cdot 2x^3 y^4$. Apply power to each factor inside the parentheses: $(ab)^n = a^n b^n$. $$(4x^4 y)^2 = 4^2 x^{4 \cdot 2} y^{1 \cdot 2} = 16x^8 y^2$$ Multiply by $2x^3 y^4$: $$16x^8 y^2 \cdot 2x^3 y^4 = (16 \cdot 2) x^{8+3} y^{2+4} = 32x^{11} y^6$$ 3. Simplify $\frac{36x^9 y^4}{4x^7 y^3}$. Divide coefficients and subtract exponents: $$\frac{36}{4} = 9$$ $$x^{9-7} = x^2$$ $$y^{4-3} = y^1 = y$$ So, $$9x^2 y$$ 4. Simplify $\frac{((2xy)^5)^3}{2x^3 y^8}$. First, simplify the numerator: $$((2xy)^5)^3 = (2xy)^{5 \cdot 3} = (2xy)^{15} = 2^{15} x^{15} y^{15}$$ Divide by denominator: $$\frac{2^{15} x^{15} y^{15}}{2 x^3 y^8} = 2^{15-1} x^{15-3} y^{15-8} = 2^{14} x^{12} y^7$$ 5. Simplify $(-5x^6 y^2)^2 - 12x^{12} y^4$. Square the first term: $$(-5)^2 x^{6 \cdot 2} y^{2 \cdot 2} = 25 x^{12} y^4$$ Subtract: $$25 x^{12} y^4 - 12 x^{12} y^4 = (25 - 12) x^{12} y^4 = 13 x^{12} y^4$$ 6. Simplify $\frac{6x^{10} y^4}{3x^8 y^7}$. Divide coefficients and subtract exponents: $$\frac{6}{3} = 2$$ $$x^{10-8} = x^2$$ $$y^{4-7} = y^{-3}$$ So, $$2 x^2 y^{-3}$$ 7. Simplify $6x^{-1} y^{-5} \cdot 4x^{-4} y^{-2}$. Multiply coefficients and add exponents: $$6 \cdot 4 = 24$$ $$x^{-1 + (-4)} = x^{-5}$$ $$y^{-5 + (-2)} = y^{-7}$$ So, $$24 x^{-5} y^{-7}$$ 8. Simplify $(3x^{-6} y^2)^3 \cdot 2x^{10} y^{-7}$. First, cube the first term: $$3^3 x^{-6 \cdot 3} y^{2 \cdot 3} = 27 x^{-18} y^6$$ Multiply by second term: $$27 x^{-18} y^6 \cdot 2 x^{10} y^{-7} = 54 x^{-18 + 10} y^{6 - 7} = 54 x^{-8} y^{-1}$$ 9. Simplify $\frac{(-2xy)^2 \cdot 10x^3 y^{11}}{8x^{10} y^4}$. Calculate numerator: $$(-2)^2 x^{2} y^{2} \cdot 10 x^3 y^{11} = 4 x^2 y^2 \cdot 10 x^3 y^{11} = 40 x^{2+3} y^{2+11} = 40 x^5 y^{13}$$ Divide by denominator: $$\frac{40 x^5 y^{13}}{8 x^{10} y^4} = \frac{40}{8} x^{5-10} y^{13-4} = 5 x^{-5} y^9$$ 10. Simplify $\frac{8x^3 \cdot 12x y^7}{3x^2 y^4} - 15 x^2 y^3$. Multiply numerator: $$8 x^3 \cdot 12 x y^7 = 96 x^{3+1} y^7 = 96 x^4 y^7$$ Divide by denominator: $$\frac{96 x^4 y^7}{3 x^2 y^4} = \frac{96}{3} x^{4-2} y^{7-4} = 32 x^2 y^3$$ Subtract $15 x^2 y^3$: $$32 x^2 y^3 - 15 x^2 y^3 = (32 - 15) x^2 y^3 = 17 x^2 y^3$$ --- Now match with Column 2: 1. $6x^7 y^2$ matches none exactly. 2. $32 x^{11} y^6$ matches none exactly. 3. $9 x^2 y$ matches none exactly. 4. $2^{14} x^{12} y^7$ is very large, no match. 5. $13 x^{12} y^4$ matches Light Green: $((3x^5 y^5)^3) / (3x^3 y^{11}) + 4x^{12} y^4$ (which simplifies to $13 x^{12} y^4$). 6. $2 x^2 y^{-3}$ matches Dark Green: $(34 x^{10} y^9) / (2 x^8 y^6) = 17 x^2 y^3$ (not exact, so no). 7. $24 x^{-5} y^{-7}$ matches Dark Blue: $3 x^{-4} y^9 \cdot 8 x^{-1} y^{-6} = 24 x^{-5} y^{3}$ (close but y exponent differs). 8. $54 x^{-8} y^{-1}$ matches Yellow: $(3 x^{-3} y)^3 \cdot 2 x y^{-4} = 54 x^{-8} y^{-1}$. 9. $5 x^{-5} y^9$ matches Red: $x^4 y^8 \cdot 5 x^{-9} y = 5 x^{-5} y^9$. 10. $17 x^2 y^3$ matches Light Green or Dark Green but better matches Light Green. Hence, the first problem simplified is $6 x^7 y^2$ with no exact match in Column 2.