Multiplication, Addition of Exponents
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I am thinking about an example from a GRE (graduate record exam) book
that was shown to me.
I think it was "Which is greater, x2+y2 or (x+y)2?
Here is the poop on how to think about examples like that. When in doubt – substitute
(if you can) for whole numbers. (In the original post, I had written real numbers instead of whole numbers. See the comment below about this by astute reader Randall Jones for important information about the difference that makes in this equation.)
So, try, say, "Which is greater, 52+32 or
(5+3)2?"
In the first case, 52 = 25 and 32 = 9, so it would be
25+9, which equals 34.
In the second case, you would first do the 5+3 (because parenthesis come first
in the order of operations) and get 8. Then you would square that, and get
64, which is clearly greater than 34.
Therefore (5+3)2 is greater than 52+32.
For an easy substitution you can do in your head in seconds, substitute 1s for x and for y:
= x2+y2 or (x+y)2
= 1+1 or 2 2
= 2 or 4
What if the example had been a bit different, though? What if it had been:
"Which is greater, x2*y2 or (x*y)2 (using multiplication instead
of addition)?
This article is continued at Mathmojo.com.
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Comment by RandallJones
For the first problem, you can also can solve the problem without substitution.
(x+y)^2= x^2 + 2xy + y^2 which is obviously greater than x^2 + y^2, that is if x and y are both greater than zero.
But you say to substitute any real numbers and if either x or y is negative, than the reverse would be true, x^2 + y^2 would be the larger algebraic expression.