e) \( 8 \sqrt{6}(3 \sqrt{2}-4 \sqrt{3}-2 \sqrt{6}) \) f) \( (2 \sqrt{5}-3 \sqrt{2})(\sqrt{5}+2 \sqrt{2}) \) g) \( \frac{1}{2} \sqrt{180}-\frac{6}{7} \sqrt{245}+\frac{2}{3} \sqrt{405} \)

Simplify the expression by following steps: - step0: Multiply the numbers: \(\left(2\sqrt{5}-3\sqrt{2}\right)\left(\sqrt{5}+2\sqrt{2}\right)\) - step1: Apply the distributive property: \(2\sqrt{5}\times \sqrt{5}+2\sqrt{5}\times 2\sqrt{2}-3\sqrt{2}\times \sqrt{5}-3\sqrt{2}\times 2\sqrt{2}\) - step2: Multiply the numbers: \(10+4\sqrt{10}-3\sqrt{10}-12\) - step3: Subtract the numbers: \(-2+\sqrt{10}\) Expand the expression \( 8 \sqrt{6}(3 \sqrt{2}-4 \sqrt{3}-2 \sqrt{6}) \) Simplify the expression by following steps: - step0: Multiply the numbers: \(8\sqrt{6}\times \left(3\sqrt{2}-4\sqrt{3}-2\sqrt{6}\right)\) - step1: Multiply the terms: \(\left(24\sqrt{2}-32\sqrt{3}-16\sqrt{6}\right)\sqrt{6}\) - step2: Apply the distributive property: \(24\sqrt{2}\times \sqrt{6}-32\sqrt{3}\times \sqrt{6}-16\sqrt{6}\times \sqrt{6}\) - step3: Multiply the numbers: \(48\sqrt{3}-32\sqrt{3}\times \sqrt{6}-16\sqrt{6}\times \sqrt{6}\) - step4: Multiply the numbers: \(48\sqrt{3}-96\sqrt{2}-16\sqrt{6}\times \sqrt{6}\) - step5: Multiply the numbers: \(48\sqrt{3}-96\sqrt{2}-96\) Expand the expression \( \frac{1}{2} \sqrt{180}-\frac{6}{7} \sqrt{245}+\frac{2}{3} \sqrt{405} \) Simplify the expression by following steps: - step0: Simplify: \(\frac{1}{2}\sqrt{180}-\frac{6}{7}\sqrt{245}+\frac{2}{3}\sqrt{405}\) - step1: Simplify the root: \(\frac{1}{2}\times 6\sqrt{5}-\frac{6}{7}\sqrt{245}+\frac{2}{3}\sqrt{405}\) - step2: Simplify the root: \(\frac{1}{2}\times 6\sqrt{5}-\frac{6}{7}\times 7\sqrt{5}+\frac{2}{3}\sqrt{405}\) - step3: Simplify the root: \(\frac{1}{2}\times 6\sqrt{5}-\frac{6}{7}\times 7\sqrt{5}+\frac{2}{3}\times 9\sqrt{5}\) - step4: Multiply the numbers: \(3\sqrt{5}-\frac{6}{7}\times 7\sqrt{5}+\frac{2}{3}\times 9\sqrt{5}\) - step5: Reduce the numbers: \(3\sqrt{5}-6\sqrt{5}+\frac{2}{3}\times 9\sqrt{5}\) - step6: Multiply the numbers: \(3\sqrt{5}-6\sqrt{5}+6\sqrt{5}\) - step7: Collect like terms: \(\left(3-6+6\right)\sqrt{5}\) - step8: Calculate: \(3\sqrt{5}\) Let's break down the calculations for each expression step by step. ### e) \( 8 \sqrt{6}(3 \sqrt{2}-4 \sqrt{3}-2 \sqrt{6}) \) 1. Distribute \( 8 \sqrt{6} \) across each term inside the parentheses: \[ 8 \sqrt{6} \cdot 3 \sqrt{2} - 8 \sqrt{6} \cdot 4 \sqrt{3} - 8 \sqrt{6} \cdot 2 \sqrt{6} \] 2. Calculate each term: - \( 8 \sqrt{6} \cdot 3 \sqrt{2} = 24 \sqrt{12} = 24 \cdot 2 \sqrt{3} = 48 \sqrt{3} \) - \( 8 \sqrt{6} \cdot 4 \sqrt{3} = 32 \sqrt{18} = 32 \cdot 3 \sqrt{2} = 96 \sqrt{2} \) - \( 8 \sqrt{6} \cdot 2 \sqrt{6} = 16 \cdot 6 = 96 \) 3. Combine the results: \[ 48 \sqrt{3} - 96 \sqrt{2} - 96 \] ### f) \( (2 \sqrt{5}-3 \sqrt{2})(\sqrt{5}+2 \sqrt{2}) \) 1. Use the distributive property (FOIL method): \[ 2 \sqrt{5} \cdot \sqrt{5} + 2 \sqrt{5} \cdot 2 \sqrt{2} - 3 \sqrt{2} \cdot \sqrt{5} - 3 \sqrt{2} \cdot 2 \sqrt{2} \] 2. Calculate each term: - \( 2 \sqrt{5} \cdot \sqrt{5} = 2 \cdot 5 = 10 \) - \( 2 \sqrt{5} \cdot 2 \sqrt{2} = 4 \sqrt{10} \) - \( -3 \sqrt{2} \cdot \sqrt{5} = -3 \sqrt{10} \) - \( -3 \sqrt{2} \cdot 2 \sqrt{2} = -6 \) 3. Combine the results: \[ 10 + 4 \sqrt{10} - 3 \sqrt{10} - 6 = 4 - 2 + \sqrt{10} = -2 + \sqrt{10} \] ### g) \( \frac{1}{2} \sqrt{180}-\frac{6}{7} \sqrt{245}+\frac{2}{3} \sqrt{405} \) 1. Simplify each square root: - \( \sqrt{180} = \sqrt{36 \cdot 5} = 6 \sqrt{5} \) - \( \sqrt{245} = \sqrt{49 \cdot 5} = 7 \sqrt{5} \) - \( \sqrt{405} = \sqrt{81 \cdot 5} = 9 \sqrt{5} \) 2. Substitute back into the expression: \[ \frac{1}{2} \cdot 6 \sqrt{5} - \frac{6}{7} \cdot 7 \sqrt{5} + \frac{2}{3} \cdot 9 \sqrt{5} \] 3. Calculate each term: - \( \frac{1}{2} \cdot 6 \sqrt{5} = 3 \sqrt{5} \) - \( -\frac{6}{7} \cdot 7 \sqrt{5} = -6 \sqrt{5} \) - \( \frac{2}{3} \cdot 9 \sqrt{5} = 6 \sqrt{5} \) 4. Combine the results: \[ 3 \sqrt{5} - 6 \sqrt{5} + 6 \sqrt{5} = 3 \sqrt{5} \] ### Final Results - e) \( 48 \sqrt{3} - 96 \sqrt{2} - 96 \) - f) \( -2 + \sqrt{10} \) - g) \( 3 \sqrt{5} \)