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

Let's break down the calculations using the properties of operations. (1) Starting with \(12 - 97 - 8 + 100\), we can rearrange it using the commutative property: \( (100 + 12 - 97 - 8) \) This simplifies to \(112 - 97 - 8\), which equals \(15 - 8 = 7\). (2) For \(17 - 83 + 82 - 10\), we can rearrange the terms: \( (17 + 82 - 83 - 10) \) This simplifies to \(99 - 93\), which gives us \(6\). (3) For \(3 \frac{1}{2} - 8 + \frac{1}{2} + 5\), let's convert mixed number to improper: \( \frac{7}{2} - 8 + \frac{1}{2} + 5\), which rearranges to \( \frac{7 + 1}{2} + (5 - 8) \) or \( \frac{8}{2} - 3 = 4 - 3 = 1\). (4) In \( \frac{26}{3} - \frac{1}{4} + \frac{1}{3} + 1\), we can combine using a common denominator: Combining terms gives us \( \frac{26}{3} + 1 = \frac{29}{3} \) and \( - \frac{1}{4} + \frac{1}{3} = \frac{-3 + 4}{12} = \frac{1}{12}\), so \( \frac{29}{3} + \frac{1}{12} = \frac{116 + 1}{12} = \frac{117}{12} \approx 9.75\) (5) For \(8 \times \frac{5}{7} \times \frac{1}{4} \times \frac{7}{10}\), notice the \(7\) cancels out: This leads to \(8 \times \frac{5}{4} \times \frac{1}{10} = \frac{40}{40} = 1\). (6) In \( \frac{16}{9} \times \frac{14}{3} \times \frac{9}{16} \times \frac{9}{14}\), we can cancel out the \(16s\) and \(14s\): This results in \( \frac{9 \times 9}{9} = 9\). (7) For \( \frac{1}{3}(9 - 6 + \frac{1}{3})\), we can simplify the inside first: It becomes \( \frac{1}{3}(3 + \frac{1}{3}) = \frac{1}{3} \times \frac{10}{3} = \frac{10}{9}\). (8) In \( \frac{1}{4}(8 - \frac{8}{3} - \frac{4}{3})\), we first combine: This becomes \( \frac{1}{4} \left( 8 - 4 \right) = \frac{1}{4} \times 4 = 1\). To summarize the results: (1) 7, (2) 6, (3) 1, (4) \(\frac{117}{12}\) or 9.75, (5) 1, (6) 9, (7) \(\frac{10}{9}\), (8) 1.