1. **Stating the problem:** We have several expressions to simplify and evaluate: - Solve for the greatest common divisor (GCD) of 3024 and 201. - Simplify and evaluate expressions A, B, C. - Calculate \( \frac{C}{3\sqrt{13}} \). - Understand the rectangular graph with dimensions 236m by 180m. --- 2. **Find the GCD of 3024 and 201:** Use the Euclidean algorithm: $$\gcd(3024,201)$$ Calculate: $$3024 \div 201 = 15 \text{ remainder } 9$$ So, $$\gcd(3024,201) = \gcd(201,9)$$ Next, $$201 \div 9 = 22 \text{ remainder } 3$$ So, $$\gcd(201,9) = \gcd(9,3)$$ Next, $$9 \div 3 = 3 \text{ remainder } 0$$ So, $$\gcd(9,3) = 3$$ **Answer:** \(\gcd(3024,201) = 3\) --- 3. **Simplify expression A:** Given: $$A = \frac{1}{5} \sqrt{275} + \frac{3^3}{7} \sqrt{539} - \frac{10^2}{22} \sqrt{704}$$ Simplify each term: - \(\sqrt{275} = \sqrt{25 \times 11} = 5\sqrt{11}\) - \(3^3 = 27\) - \(\sqrt{539} = \sqrt{7 \times 7 \times 11} = 7\sqrt{11}\) - \(10^2 = 100\) - \(\sqrt{704} = \sqrt{64 \times 11} = 8\sqrt{11}\) Substitute back: $$A = \frac{1}{5} \times 5\sqrt{11} + \frac{27}{7} \times 7\sqrt{11} - \frac{100}{22} \times 8\sqrt{11}$$ Simplify coefficients: $$A = \sqrt{11} + 27\sqrt{11} - \frac{800}{22} \sqrt{11}$$ Simplify \(\frac{800}{22} = \frac{400}{11}\): $$A = \sqrt{11} + 27\sqrt{11} - \frac{400}{11} \sqrt{11}$$ Combine terms: $$A = \left(1 + 27 - \frac{400}{11}\right) \sqrt{11} = \left(28 - \frac{400}{11}\right) \sqrt{11}$$ Convert 28 to fraction: $$28 = \frac{308}{11}$$ So, $$A = \left(\frac{308}{11} - \frac{400}{11}\right) \sqrt{11} = -\frac{92}{11} \sqrt{11}$$ --- 4. **Simplify expression B:** Given: $$B = (\sqrt{13} - 6)(2\sqrt{13} - 5) + 17\sqrt{13}$$ Multiply the binomials: $$(\sqrt{13})(2\sqrt{13}) = 2 \times 13 = 26$$ $$(\sqrt{13})(-5) = -5\sqrt{13}$$ $$(-6)(2\sqrt{13}) = -12\sqrt{13}$$ $$(-6)(-5) = 30$$ Sum these: $$26 - 5\sqrt{13} - 12\sqrt{13} + 30 = (26 + 30) - (5\sqrt{13} + 12\sqrt{13}) = 56 - 17\sqrt{13}$$ Add \(17\sqrt{13}\): $$B = 56 - 17\sqrt{13} + 17\sqrt{13} = 56$$ --- 5. **Simplify expression C:** Given: $$C = \sqrt{3}(\sqrt{12} + 4\sqrt{3}) - 5$$ Simplify inside the parentheses: $$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$$ So, $$C = \sqrt{3}(2\sqrt{3} + 4\sqrt{3}) - 5 = \sqrt{3} \times 6\sqrt{3} - 5$$ Multiply: $$\sqrt{3} \times \sqrt{3} = 3$$ So, $$C = 6 \times 3 - 5 = 18 - 5 = 13$$ --- 6. **Calculate \( \frac{C}{3\sqrt{13}} \):** Substitute \(C = 13\): $$\frac{13}{3\sqrt{13}}$$ Rationalize the denominator: $$\frac{13}{3\sqrt{13}} \times \frac{\sqrt{13}}{\sqrt{13}} = \frac{13\sqrt{13}}{3 \times 13} = \frac{\sqrt{13}}{3}$$ --- 7. **Rectangular graph interpretation:** The rectangle has width 236m and height 180m. - Area of the rectangle: $$\text{Area} = 236 \times 180 = 42480 \text{ m}^2$$ - This could represent a field or plot area. --- **Final answers:** - \(\gcd(3024,201) = 3\) - \(A = -\frac{92}{11} \sqrt{11}\) - \(B = 56\) - \(C = 13\) - \(\frac{C}{3\sqrt{13}} = \frac{\sqrt{13}}{3}\) - Area of rectangle = 42480 m\(^2\)