Simplify. \( \begin{array}{ll}\text { 25. }(7 \sqrt{2}-3 \sqrt{3})(4 \sqrt{6}+3 \sqrt{12}) & \text { 26. }(8 \sqrt{5}-6 \sqrt{3})(8 \sqrt{5}+6 \sqrt{3}) \\ \text { 27. }(12 \sqrt{10}-6 \sqrt{5})(12 \sqrt{10}+6 \sqrt{5}) & \text { 28. }(6 \sqrt{3}+5 \sqrt{2})(2 \sqrt{6}+3 \sqrt{8}) \\ 36 \sqrt{2}+36 \sqrt{10}+20\end{array} \) Operations with Radical Expressions

Simplify the expression by following steps: - step0: Multiply the terms: \(\left(8\sqrt{5}-6\sqrt{3}\right)\left(8\sqrt{5}+6\sqrt{3}\right)\) - step1: Simplify the product: \(\left(8\sqrt{5}\right)^{2}-\left(6\sqrt{3}\right)^{2}\) - step2: Evaluate the power: \(320-108\) - step3: Subtract the numbers: \(212\) Expand the expression \( (12 \sqrt{10}-6 \sqrt{5})(12 \sqrt{10}+6 \sqrt{5}) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(12\sqrt{10}-6\sqrt{5}\right)\left(12\sqrt{10}+6\sqrt{5}\right)\) - step1: Simplify the product: \(\left(12\sqrt{10}\right)^{2}-\left(6\sqrt{5}\right)^{2}\) - step2: Evaluate the power: \(1440-180\) - step3: Subtract the numbers: \(1260\) Expand the expression \( (6 \sqrt{3}+5 \sqrt{2})(2 \sqrt{6}+3 \sqrt{8}) \) Simplify the expression by following steps: - step0: Simplify: \(\left(6\sqrt{3}+5\sqrt{2}\right)\left(2\sqrt{6}+3\sqrt{8}\right)\) - step1: Simplify the root: \(\left(6\sqrt{3}+5\sqrt{2}\right)\left(2\sqrt{6}+3\times 2\sqrt{2}\right)\) - step2: Multiply the terms: \(\left(6\sqrt{3}+5\sqrt{2}\right)\left(2\sqrt{6}+6\sqrt{2}\right)\) - step3: Apply the distributive property: \(6\sqrt{3}\times 2\sqrt{6}+6\sqrt{3}\times 6\sqrt{2}+5\sqrt{2}\times 2\sqrt{6}+5\sqrt{2}\times 6\sqrt{2}\) - step4: Multiply the numbers: \(36\sqrt{2}+36\sqrt{6}+20\sqrt{3}+60\) Expand the expression \( (7 \sqrt{2}-3 \sqrt{3})(4 \sqrt{6}+3 \sqrt{12}) \) Simplify the expression by following steps: - step0: Simplify: \(\left(7\sqrt{2}-3\sqrt{3}\right)\left(4\sqrt{6}+3\sqrt{12}\right)\) - step1: Simplify the root: \(\left(7\sqrt{2}-3\sqrt{3}\right)\left(4\sqrt{6}+3\times 2\sqrt{3}\right)\) - step2: Multiply the terms: \(\left(7\sqrt{2}-3\sqrt{3}\right)\left(4\sqrt{6}+6\sqrt{3}\right)\) - step3: Apply the distributive property: \(7\sqrt{2}\times 4\sqrt{6}+7\sqrt{2}\times 6\sqrt{3}-3\sqrt{3}\times 4\sqrt{6}-3\sqrt{3}\times 6\sqrt{3}\) - step4: Multiply the numbers: \(56\sqrt{3}+42\sqrt{6}-36\sqrt{2}-54\) Let's simplify each expression step by step. ### 25. \( (7 \sqrt{2}-3 \sqrt{3})(4 \sqrt{6}+3 \sqrt{12}) \) Using the distributive property (FOIL method): \[ (7 \sqrt{2})(4 \sqrt{6}) + (7 \sqrt{2})(3 \sqrt{12}) - (3 \sqrt{3})(4 \sqrt{6}) - (3 \sqrt{3})(3 \sqrt{12}) \] Calculating each term: 1. \( 7 \sqrt{2} \cdot 4 \sqrt{6} = 28 \sqrt{12} = 28 \cdot 2 \sqrt{3} = 56 \sqrt{3} \) 2. \( 7 \sqrt{2} \cdot 3 \sqrt{12} = 21 \sqrt{24} = 21 \cdot 2 \sqrt{6} = 42 \sqrt{6} \) 3. \( -3 \sqrt{3} \cdot 4 \sqrt{6} = -12 \sqrt{18} = -12 \cdot 3 \sqrt{2} = -36 \sqrt{2} \) 4. \( -3 \sqrt{3} \cdot 3 \sqrt{12} = -9 \sqrt{36} = -9 \cdot 6 = -54 \) Combining all terms: \[ 56 \sqrt{3} + 42 \sqrt{6} - 36 \sqrt{2} - 54 \] ### 26. \( (8 \sqrt{5}-6 \sqrt{3})(8 \sqrt{5}+6 \sqrt{3}) \) This is a difference of squares: \[ (8 \sqrt{5})^2 - (6 \sqrt{3})^2 = 64 \cdot 5 - 36 \cdot 3 = 320 - 108 = 212 \] ### 27. \( (12 \sqrt{10}-6 \sqrt{5})(12 \sqrt{10}+6 \sqrt{5}) \) Again, this is a difference of squares: \[ (12 \sqrt{10})^2 - (6 \sqrt{5})^2 = 144 \cdot 10 - 36 \cdot 5 = 1440 - 180 = 1260 \] ### 28. \( (6 \sqrt{3}+5 \sqrt{2})(2 \sqrt{6}+3 \sqrt{8}) \) Using the distributive property: \[ (6 \sqrt{3})(2 \sqrt{6}) + (6 \sqrt{3})(3 \sqrt{8}) + (5 \sqrt{2})(2 \sqrt{6}) + (5 \sqrt{2})(3 \sqrt{8}) \] Calculating each term: 1. \( 6 \sqrt{3} \cdot 2 \sqrt{6} = 12 \sqrt{18} = 12 \cdot 3 \sqrt{2} = 36 \sqrt{2} \) 2. \( 6 \sqrt{3} \cdot 3 \sqrt{8} = 18 \sqrt{24} = 18 \cdot 2 \sqrt{6} = 36 \sqrt{6} \) 3. \( 5 \sqrt{2} \cdot 2 \sqrt{6} = 10 \sqrt{12} = 10 \cdot 2 \sqrt{3} = 20 \sqrt{3} \) 4. \( 5 \sqrt{2} \cdot 3 \sqrt{8} = 15 \sqrt{16} = 15 \cdot 4 = 60 \) Combining all terms: \[ 36 \sqrt{2} + 36 \sqrt{6} + 20 \sqrt{3} + 60 \] ### Summary of Results - **25:** \( 56 \sqrt{3} + 42 \sqrt{6} - 36 \sqrt{2} - 54 \) - **26:** \( 212 \) - **27:** \( 1260 \) - **28:** \( 36 \sqrt{2} + 36 \sqrt{6} + 20 \sqrt{3} + 60 \)