By using the commutative, associative, or distributive property, calculate the following: \( \begin{array}{ll}\text { (1) } 12-97-8+100 & \text { (2) } 17-83+82-10 \\ \text { (3) } 3 \frac{1}{2}-8+\frac{1}{2}+5 & \text { (4) } \frac{26}{3}-\frac{1}{4}+\frac{1}{3}+1 \\ \text { (5) } 8 \times \frac{5}{7} \times \frac{1}{4} \times \frac{7}{10} & \text { (6) } \frac{16}{9} \times \frac{14}{3} \times \frac{9}{16} \times \frac{9}{14} \\ \text { (7) } \frac{1}{3}\left(9-6+\frac{1}{3}\right) & \text { (8) } \frac{1}{4}\left(8-\frac{8}{3}-\frac{4}{3}\right)\end{array} \)

Calculate the value by following steps: - step0: Calculate: \(12-97-8+100\) - step1: Calculate: \(7\) Calculate or simplify the expression \( 17-83+82-10 \). Calculate the value by following steps: - step0: Calculate: \(17-83+82-10\) - step1: Calculate: \(6\) Calculate or simplify the expression \( 3.5-8+0.5+5 \). Calculate the value by following steps: - step0: Calculate: \(3.5-8+0.5+5\) - step1: Calculate: \(1\) Calculate or simplify the expression \( 8*(5/7)*(1/4)*(7/10) \). Calculate the value by following steps: - step0: Calculate: \(8\times \frac{5}{7}\times \frac{1}{4}\times \frac{7}{10}\) - step1: Reduce the fraction: \(8\times 5\times \frac{1}{4}\times \frac{1}{10}\) - step2: Multiply the terms: \(40\times \frac{1}{4}\times \frac{1}{10}\) - step3: Multiply the terms: \(10\times \frac{1}{10}\) - step4: Reduce the fraction: \(1\times 1\) - step5: Multiply the terms: \(1\) Calculate or simplify the expression \( (16/9)*(14/3)*(9/16)*(9/14) \). Calculate the value by following steps: - step0: Calculate: \(\frac{16}{9}\times \frac{14}{3}\times \frac{9}{16}\times \frac{9}{14}\) - step1: Reduce the fraction: \(\frac{1}{9}\times \frac{14}{3}\times 9\times \frac{9}{14}\) - step2: Reduce the fraction: \(1\times \frac{14}{3}\times 1\times \frac{9}{14}\) - step3: Reduce the fraction: \(1\times \frac{1}{3}\times 1\times 9\) - step4: Multiply the terms: \(\frac{1}{3}\times 1\times 9\) - step5: Multiply the terms: \(\frac{1}{3}\times 9\) - step6: Reduce the numbers: \(1\times 3\) - step7: Simplify: \(3\) Calculate or simplify the expression \( (1/4)*(8-8/3-4/3) \). Calculate the value by following steps: - step0: Calculate: \(\frac{1}{4}\left(8-\frac{8}{3}-\frac{4}{3}\right)\) - step1: Subtract the numbers: \(\frac{1}{4}\times 4\) - step2: Reduce the numbers: \(1\times 1\) - step3: Simplify: \(1\) Calculate or simplify the expression \( (1/3)*(9-6+1/3) \). Calculate the value by following steps: - step0: Calculate: \(\frac{1}{3}\left(9-6+\frac{1}{3}\right)\) - step1: Calculate: \(\frac{1}{3}\times \frac{10}{3}\) - step2: Multiply the fractions: \(\frac{10}{3\times 3}\) - step3: Multiply: \(\frac{10}{9}\) Calculate or simplify the expression \( 26/3-1/4+1/3+1 \). Calculate the value by following steps: - step0: Calculate: \(\frac{26}{3}-\frac{1}{4}+\frac{1}{3}+1\) - step1: Reduce fractions to a common denominator: \(\frac{26\times 4}{3\times 4}-\frac{3}{4\times 3}+\frac{4}{3\times 4}+\frac{3\times 4}{3\times 4}\) - step2: Multiply the numbers: \(\frac{26\times 4}{12}-\frac{3}{4\times 3}+\frac{4}{3\times 4}+\frac{3\times 4}{3\times 4}\) - step3: Multiply the numbers: \(\frac{26\times 4}{12}-\frac{3}{12}+\frac{4}{3\times 4}+\frac{3\times 4}{3\times 4}\) - step4: Multiply the numbers: \(\frac{26\times 4}{12}-\frac{3}{12}+\frac{4}{12}+\frac{3\times 4}{3\times 4}\) - step5: Multiply the numbers: \(\frac{26\times 4}{12}-\frac{3}{12}+\frac{4}{12}+\frac{3\times 4}{12}\) - step6: Transform the expression: \(\frac{26\times 4-3+4+3\times 4}{12}\) - step7: Multiply the numbers: \(\frac{104-3+4+3\times 4}{12}\) - step8: Multiply the numbers: \(\frac{104-3+4+12}{12}\) - step9: Calculate: \(\frac{117}{12}\) - step10: Reduce the fraction: \(\frac{39}{4}\) Let's calculate each expression step by step using the properties of arithmetic. 1. **Expression:** \( 12 - 97 - 8 + 100 \) - Rearranging using the associative property: \[ (12 + 100) - (97 + 8) = 112 - 105 = 7 \] **Result:** \( 7 \) 2. **Expression:** \( 17 - 83 + 82 - 10 \) - Rearranging using the associative property: \[ (17 + 82) - (83 + 10) = 99 - 93 = 6 \] **Result:** \( 6 \) 3. **Expression:** \( 3 \frac{1}{2} - 8 + \frac{1}{2} + 5 \) - Converting mixed number to improper fraction: \[ \frac{7}{2} - 8 + \frac{1}{2} + 5 = \frac{7}{2} - \frac{16}{2} + \frac{1}{2} + \frac{10}{2} \] - Combining: \[ \frac{7 - 16 + 1 + 10}{2} = \frac{2}{2} = 1 \] **Result:** \( 1 \) 4. **Expression:** \( \frac{26}{3} - \frac{1}{4} + \frac{1}{3} + 1 \) - Finding a common denominator (12): \[ \frac{104}{12} - \frac{3}{12} + \frac{4}{12} + \frac{12}{12} = \frac{104 - 3 + 4 + 12}{12} = \frac{117}{12} = \frac{39}{4} \text{ or } 9 \frac{3}{4} \text{ or } 9.75 \] **Result:** \( \frac{39}{4} \) 5. **Expression:** \( 8 \times \frac{5}{7} \times \frac{1}{4} \times \frac{7}{10} \) - Rearranging and simplifying: \[ 8 \times \frac{5 \times 7}{7 \times 4 \times 10} = 8 \times \frac{5}{40} = 1 \] **Result:** \( 1 \) 6. **Expression:** \( \frac{16}{9} \times \frac{14}{3} \times \frac{9}{16} \times \frac{9}{14} \) - Rearranging and simplifying: \[ \frac{16 \times 14 \times 9 \times 9}{9 \times 16 \times 3 \times 14} = \frac{81}{27} = 3 \] **Result:** \( 3 \) 7. **Expression:** \( \frac{1}{3}(9 - 6 + \frac{1}{3}) \) - Simplifying inside the parentheses: \[ 9 - 6 + \frac{1}{3} = 3 + \frac{1}{3} = \frac{9}{3} + \frac{1}{3} = \frac{10}{3} \] - Now multiplying: \[ \frac{1}{3} \times \frac{10}{3} = \frac{10}{9} \text{ or } 1 \frac{1}{9} \text{ or } 1.1\dot{1} \] **Result:** \( \frac{10}{9} \) 8. **Expression:** \( \frac{1}{4}(8 - \frac{8}{3} - \frac{4}{3}) \) - Finding a common denominator (3): \[ 8 = \frac{24}{3}, \text{ so } \frac{24}{3} - \frac{8}{3} - \frac{4}{3} = \frac{24 - 8 - 4}{3} = \frac{12}{3} = 4 \] - Now multiplying: \[ \frac{1}{4} \times 4 = 1 \] **Result:** \( 1 \) ### Summary of Results: 1. \( 7 \) 2. \( 6 \) 3. \( 1 \) 4. \( \frac{39}{4} \) 5. \( 1 \) 6. \( 3 \) 7. \( \frac{10}{9} \) 8. \( 1 \)