What Does It Mean to DistributeDate: 08/25/2007 at 22:59:01 From: Andrea Subject: distributive property I understand the basics of the distributive property but I don't understand all the different parts of it, like there's the distributive property of multiplication over addition and over subtraction and the distributive property of division over addition. And something with square roots distributing over multiplication and over addition. I have a worksheet and I have no idea. I just don't understand all the different parts of the property. What does something distributing over something else mean? (I.e. division distributing over addition.) Date: 08/25/2007 at 23:50:22 From: Doctor Peterson Subject: Re: distributive property Hi, Andrea. I wouldn't call these different "parts" of one property; they are equivalent properties of different operations. In general, we say that one operation, "@", distributes over another, "#", if a @ (b # c) = (a @ b) # (a @ c) for all a, b, c and, on the other side, (a # b) @ c = (a @ c) # (b @ c) That is, doing @ on the result of # gives the same result as doing @ first to EACH element and then combining them with #. The idea of distribution is just like "distributing" money to a group of people: we give the same amount to EACH of them. Now, we have to consider each pair of operations separately; some operations distribute over others, and some don't. To test it, write the statements above using the given operations, and decide whether each is always true. Among the pairs that do have this property, as you mentioned, are multiplication over addition: a*(b + c) = a*b + a*c (a + b)*c = a*c + b*c multiplication over subtraction: a*(b - c) = a*b - a*c (a - b)*c = a*c - b*c Among those that DON'T distribute (on the left) are division over addition: a/(b + c) = a/b + a/c -- no exponentiation over multiplication: a^(b * c) = a^b * a^c -- no exponentiation over division: a^(b / c) = a^b / a^c -- no But these do distribute on the right: division over addition: (a + b)/c = a/c + b/c exponentiation over multiplication: (a * b)^c = a^c * b^c exponentiation over division: (a / b)^c = a^c / b^c We don't usually bother talking about this property for operations that are not commutative, because this sort of detail is confusing. Since a square root is really the 1/2 power, you can talk about it distributing: root over multiplication: (a * b)^(1/2) = a^(1/2) * b^(1/2) sqrt(a * b) = sqrt(a) * sqrt(b) If your assignment is to test different combinations of operations and see if they distribute, I've probably answered a lot of them; the non-commutative operations are messy, and I'm not sure how you'd be expected to handle them. It might help if you show me the actual wording of the assignment. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 08/26/2007 at 00:58:54 From: Andrea Subject: Thank you (distributive property) Thank you so much - you've made my day! And I totally get it now! This makes so much more sense! Thank you so so so...much! |
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