Square RootDate: 03/15/99 at 09:45:55 From: Sean Mooney Subject: Square Root Why cannot you take the square root of a sum or difference separately - i.e. why is sqrt(9+4) not equal to sqrt(9) + sqrt(4)? Date: 03/15/99 at 17:48:35 From: Doctor Rick Subject: Re: Square Root You want to know why the square root of 13 is not sqrt(9) + sqrt(4), which is 3+2 = 5. If 5 were the square root of 13, then we could square 5 and we would get 13. Do we? No, we get 25. So clearly it does not work. But why not? Let us look into this a little more closely. We will square (sqrt(9) + sqrt(4)) using FOIL (or the distributive property): (sqrt(9) + sqrt(4))^2 = sqrt(9)^2 + 2*sqrt(9)*sqrt(4) + sqrt(4)^2 = 9 + 2*sqrt(36) + 4 You see that we do get 9 and 4, but we also get a "cross" term, 2*sqrt(36). It is this cross term that messes up the pattern. What you are asking about, or perhaps wishing for, is a sort of distributive property of square root over addition, that works the same way the usual distributive property (of multiplication over addition) works: c(a+b) = c*a + c*b (TRUE - distributive property) sqrt(a+b) = sqrt(a) + sqrt(b) (FALSE) You can distribute a product over the members of a sum, but you cannot distribute a square root over the members of a sum. However, you CAN distribute a square root over the members of a PRODUCT: sqrt(a*b) = sqrt(a) * sqrt(b) (TRUE) This is a special case of a distributive property of POWERS over MULTIPLICATION. n n n (a*b) = a * b If n = 1/2, this is the same as above. You can think in terms of a hierarchy of operations: powers/roots multiplication/division addition/subtraction Think about this: powers are to multiplication as multiplication is to addition. When you first learned multiplication, it was defined as adding a number to itself a certain number of times: 3 * 5 = 5 + 5 + 5. In the same way, powers were first defined as multiplying a number by itself a certain number of times: 5^3 = 5 * 5 * 5. When you left the realm of integers, these simple definitions were no longer enough; but logarithms put the analogy on a solid mathematical footing for real numbers, transforming multiplication into addition and powers into multiplication. The distributive property of powers over multiplication follows from the definition of powers in the same way that the distributive property of multiplication over addition follows from the definition of multiplication. But there is no comparable way of deriving a property for powers and addition; you cannot jump 2 steps in the hierarchy and get a distributive property. I hope this answers your question, but I also hope it stirs up more questions in your mind. There are a lot of interrelationships in the world of numbers waiting to be explored. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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