Question 156679
Hi, Hope I can help
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I have to do a project about the associative property. I need example problems but I don't quite understand it well enough. I already have some simple problems but my teacher wants challenge problems too. Can you give me an example of a challenge problem for the associative property of addition and another for the associative property of multiplication so I can see how to make one. Thank you very much. 
Troy
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I think of associative property as " how you arrange the parentheses, associative = "arrange"(parentheses),(both associative, and arrange start with "a" )
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Here are some examples of associative property
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Example: {{{ (5 + 3) + 7 = 5 + (3 + 7) }}} = {{{ (8) + 7 = 5 + (10) }}} = {{{ 15 = 15 }}}
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Example: {{{ (6 - 4) + 3 = 6 + (-4+3) }}} = {{{ (2) + 3 = 6 + (-1) }}} = {{{ 5 = 6 -1 }}} = {{{ 5 = 5 }}} ( {{{ 6-4 }}} is the same thing as saying {{{ 6 + (-4) }}})
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Example: {{{ (x + 8x) + 10x = x + (8x + 10x) }}} = {{{ (9x) + 10x = x + (18x) }}} = {{{ 19x = 19x }}}
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Now lets use multiplication in the associative property
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Example: {{{ (2 * 6) * 7 = 2 * (6 * 7) }}} = {{{ (12) * 7 = 2 * (42) }}} = {{{ 84 = 84 }}}
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Example: {{{ (x * 4) * y = x * (4 * y) }}} = {{{ (4x) * y = x * (4y) }}} = {{{ 4xy = 4xy }}}
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Example: {{{ (x^2 * 5) * y^3 = x^2 * (5 * y^3) }}} = {{{ (5x^2) * y^3 = x^2 * (5y^3) }}} = {{{ 5x^2y^3 = 5x^2y^3 }}}
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Here is a challenging one for multiplication
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{{{ (a^12 * (12b)^3)* (8c)^4 = a^12 * ( (12b)^16 * (8c)^10) }}} =
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 {{{ (a^12 * 1728b^3)* 4096c^4 = a^12 * ( 1728b^3 * 4096c^4) }}} =
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 {{{ (1728a^12b^3)* 4096c^4 = a^12 * ( 7077888b^3c^4) }}} = 
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{{{ 7077888a^12b^3c^4 = 7077888a^12b^3c^4 }}}
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If you are trying to use just numbers, here is another multiplication using associative property
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{{{ (((6)(4))^2 * ((3)(12))^3) * ((5)(4))^4 = ((6)(4))^2 * (((3)(12))^3 * ((5)(4))^4) }}} =
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{{{ ((24)^2 * (36)^3) * (20)^4 = (24)^2 * ((36)^3 * (20)^4) }}} =
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{{{ (576 * 46656) * 160000 = 576 * (46656 * 160000) }}} =
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{{{ (26873856) * 160000 = 576 * (7464960000) }}} =
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{{{ 4299816960000 = 4299816960000 }}}
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You can always use smaller numbers, if that was too hard
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A challenging problem using addition and associative problems is
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{{{ (((8)(7))^2 + ((5)(3))^3) + ((4)(9))^3 = ((8)(7))^2 + (((5)(3))^3 + ((4)(9))^3) }}} =
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{{{ ((56)^2 + (15)^3) + (36)^3 = (56)^2 + ((15)^3 + (36)^3) }}} = 
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{{{ (3136 + 3375) + 46656 = 3136 + (3375 + 46656) }}}
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{{{ (6511) + 46656 = 3136 + (50031) }}} =
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{{{ 53167 = 53167 }}}
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You can also do more than three numbers
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{{{ (1+2)+3+4+5+6+7+8 = 1+(2+3)+4+5+6+7+8 = 1+2+3+4+5+(6+7)+8 }}} =
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{{{ (3)+3+4+5+6+7+8 = 1+(5)+4+5+6+7+8 = 1+2+3+4+5+(13)+8 }}} =
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{{{ (3)+33 = (5) + 31 = (13)+23 }}} =
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{{{ 36 = 36 = 36 }}}
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You can also do more than two numbers in side the parentheses
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Lets use multiplication
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{{{ (2*3*5)*7*11*13*15*20 = 2*3*5*(7*11*13)*15*20 = 2*3*5*7*(11*13*15)*20 }}} =
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{{{ (30)*7*11*13*15*20 = 2*3*5*(1001)*15*20 = 2*3*5*7*(2145)*20 }}} =
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{{{ (30)* 300300 = (1001)*9000 = (2145)* 4200 }}}
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{{{ 9009000= 9009000= 9009000}}}
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Hope I helped, Levi