The length of a rectangle is \( \sqrt{3}+1 \) and its breadth is \( \sqrt{3}-1 \). Determine the length of its diagonal in surd form. Show that \( \frac{\sqrt{12}-\sqrt{75}}{3^{\frac{3}{2}}}=-1 \) without using a calculator.

Let's tackle each part of the problem step by step. --- ### **1. Determining the Diagonal of the Rectangle** Given: - **Length (\( l \))** = \( \sqrt{3} + 1 \) - **Breadth (\( b \))** = \( \sqrt{3} - 1 \) **Objective:** Find the length of the diagonal (\( d \)) in surd form. **Method:** The diagonal of a rectangle can be found using the Pythagorean theorem: \[ d = \sqrt{l^2 + b^2} \] **Calculations:** 1. **Calculate \( l^2 \):** \[ l^2 = (\sqrt{3} + 1)^2 = (\sqrt{3})^2 + 2 \times \sqrt{3} \times 1 + 1^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3} \] 2. **Calculate \( b^2 \):** \[ b^2 = (\sqrt{3} - 1)^2 = (\sqrt{3})^2 - 2 \times \sqrt{3} \times 1 + 1^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3} \] 3. **Sum \( l^2 + b^2 \):** \[ l^2 + b^2 = (4 + 2\sqrt{3}) + (4 - 2\sqrt{3}) = 8 \] 4. **Determine \( d \):** \[ d = \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \] **Conclusion:** \[ \boxed{2\sqrt{2}} \] --- ### **2. Proving \( \frac{\sqrt{12} - \sqrt{75}}{3^{\frac{3}{2}}} = -1 \)** **Objective:** Simplify the expression without using a calculator to show that it equals \(-1\). **Method:** 1. **Simplify the Square Roots:** - \( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \) - \( \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} \) 2. **Substitute Back into the Expression:** \[ \frac{\sqrt{12} - \sqrt{75}}{3^{\frac{3}{2}}} = \frac{2\sqrt{3} - 5\sqrt{3}}{3^{\frac{3}{2}}} \] \[ = \frac{-3\sqrt{3}}{3^{\frac{3}{2}}} \] 3. **Simplify the Denominator:** - \( 3^{\frac{3}{2}} = 3^{1} \times 3^{\frac{1}{2}} = 3 \times \sqrt{3} = 3\sqrt{3} \) 4. **Final Simplification:** \[ \frac{-3\sqrt{3}}{3\sqrt{3}} = \frac{-3}{3} = -1 \] **Conclusion:** \[ \boxed{-1} \] --- Both parts of the problem have been addressed: 1. The length of the diagonal is \( \boxed{2\sqrt{2}} \). 2. The expression simplifies to \( \boxed{-1} \).