Section 1.6 Properties of Real Numbers and Simplifying Expressions 109 95. 96. 97. 98. 1 2 a 4b For Exercises 99–126, simplify by clearing parentheses and combining like terms. (See Examples 10–12.) 99. 100. 101. 312x 42 10 214a 32 14 41w 32 12 4w 512r 62 30 10r 5 31x 42 4 213x 82 102. 103. 104. 312t 4w2 812t 4w2 515y 9z2 313y 6z2 21q 5u2 12q 8u2 105. 106. 107. 3 4 18 4q2 7 108. 109. 110. 1015.1a 3.12 4 10013.14p 1.052 212 4m 21m 32 2m 6 111. 112. 113. 1 10110p 52 1 2110q 22 114. 115. 116. 7n 21n 32 6 n 6n 8k 41k 12 7 k 61x 32 12 41x 32 117. 118. 119. 51y 42 3 61y 72 0.216c 1.62 c 1.115 8x2 3.1 120. 121. 122. 2.2c 0.32 8.8x 8.6 6 238 312x 424 10x 3 533 41y 224 8y 123. 124. 125. 126. Expanding Your Skills For Exercises 127–134, determine if the expressions are equivalent. If two expressions are not equivalent, state why. 127. 3a b, b 3a 128. 4y 1, 1 4y 129. 2c 7, 9c 130. 5z 4, 9z 131. 132. 133. 134. 5x 3, 3 5x 6d 7, 7 6d 5x 3, 3 5x 8 2x, 2x 8 135. Which grouping of terms is easier to compute, ? 1518 1825 2 13 5 or 518 1213 23 2 136. Which grouping of terms is easier to compute, ? 11825 135 2 137. As a small child in school, the great mathematician Karl Friedrich Gauss (1777–1855) was said to have found the sum of the integers from 1 to 100 mentally: 1 2 3 4 p 99 100 Rather than adding the numbers sequentially, he added the numbers in pairs: 11 992 12 982 13 972 p 100 a. Use this technique to add the integers from 1 to 10. 55 1 2 3 4 5 6 7 8 9 10 b. Use this technique to add the integers from 1 to 20. 210 11825 135 2 11427 213 2 2 3 or 1427 1213 23 2 1 3321z 12 51z 224 1 63312t 22 81t 224 1 5115 4p2 1 33b 41b 22 8b 12 3q2 1 361x 3y2 16x 5y2 23y 16t 92 10 2.8z 8.1z 6 15.2 2 5 2t 3 5 t 6 5 1 4 a b 3 4 8x3y 5xy 3 7 6xy x3y a 5b 7x3y xy 4 5.3z 9.2 3t 7 5 6x 22 8a 20 3x 17 6x 12 10t 44w 16y 27z 18u 2t 7 3q 1 51a 27 314p 107 7b 8 4q 1 3 9 5 p 5 2 3k 11 2x 18 y 25 2x 34 9z 35 28y 58 12t 61 Not equivalent. The terms are not like terms and cannot be combined. Not equivalent. The terms are not like terms and cannot be combined. Equivalent Equivalent Not equivalent; subtraction Not equivalent; subtraction Equivalent Equivalent is not commutative. is not commutative. is easier. 1427 is easier. 518 Writing Translating Expression Geometry Scientific Calculator Video
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