My Math Forum Order of Operations comprehension question

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 September 23rd, 2017, 02:31 AM #1 Newbie   Joined: Sep 2017 From: UK Posts: 1 Thanks: 0 Order of Operations comprehension question Hello, this is my first post here, so please go easy on me! Thank you for your time. As far as I understand: if $\displaystyle x = a + b$ then $\displaystyle x^{2} = (a + b)^{2}$ not $\displaystyle x^{2} = a^{2} + b^{2}$ which I can easily check by asking does $\displaystyle 5^{2} = 3^{2} + 2^{2}$? No. My question is does anyone have a way of explaining this property so that I might better understand *why* this property exists, beyond simply memorising it as a property of algebra? The reason I ask is because sometimes I make this mistake in algebraic manipulations; a specific example would be from physics: since $\displaystyle v = u + at$ $\displaystyle v^{2} = (u + at)^{2} = u^{2} + 2at + a^{2}t^{2}$ However, if I am not careful, in practise I have previously resolved this as: $\displaystyle v^{2} = u^{2} + a^{2}t^{2}$, since in my mind I am "just squaring both sides." So my request is if anyone could give me an understanding to help cement this algebraic rule in my mind in a deeper way, so I don't make this mistake again. I hope that's clear. Thanks in advance for any insight. Sam
 October 1st, 2017, 02:15 AM #2 Global Moderator   Joined: Dec 2006 Posts: 18,243 Thanks: 1439 $(a + b)^2 = (a + b)(a + b) = a(a + b) + b(a + b) = a*a + ab + b*a + b*b = a^2 + 2ab + b^2$
 October 1st, 2017, 02:20 AM #3 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,453 Thanks: 489 Math Focus: Yet to find out. This mistake has a name of its own! See here. https://en.m.wikipedia.org/wiki/Freshman%27s_dream
October 1st, 2017, 04:12 AM   #4
Math Team

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Quote:
 Originally Posted by sdparsons since $\displaystyle v = u + at$ $\displaystyle v^{2} = (u + at)^{2} = u^{2} + 2at + a^{2}t^{2}$
Should be: = u^2 + 2uat + a^2t^2

Code:
 2 + 3
2 + 3
-------
6 + 9 : do bottom "3" next
-----------
4 +12 + 9 = 25
There's really 2 multiplications:
2 * (2 + 3), then 3 * (2 + 3)

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