Order of Operations and Square Roots

Date: 10/09/2009 at 16:16:52
From: Tracey
Subject: When do you do a square root in the order of ops

At what step in the order of operations (Clear parentheses, 
exponents, multiply and divide in order presented, add and subtract 
in order presented) does one work out a square root? I haven't seen 
it much yet, but it appears in expressions and equations toward the 
end of a friend's child's 6th grade math book. For instance,
14 plus (24 minus 12) squared divided by 2 times 3 cubed plus (4 
minus 2 squared) plus the square root of 9? At which step would you 
replace the square root of 9 with 3?  Thanks!

It seems like it would go with either exponents or the multiplication
and division step, but I don't know which; and I can't (for the life
of me) find an explanation in the textbook; AND I can't remember what
I learned in high school algebra . . . help!

Here goes:
14 + (24 - 12)squared divided by 2 x 3cubed + (4 - 2squared) + 
     square root of 9
14 + (12)squared divided by 2 x 3cubed + (4 - 2 squared) + square  
     root of 9
14 + 12squared divided by 2 x 3cubed + (4 - 4) + square root of 9
14 + 12squared divided by 2 x 3cubed + 0 + square root of 9
14 + 144 divided by 2 x 3cubed + 0 + square root of 9
14 + 144 divided by 2 x 27 + 0 + square root of 9
14 + 72 x 27 + 0 + square root of 9
14 + 1,944 + 0 + square root of 9
14 + 1,944 + 0 + 3
1,958 + 0 + 3
1,958 + 3
1,961

In this problem, I don't think it would change the effect to include 
the square root at the time you do exponents; however, I know that 
the order of ops is around to provide stability when it WOULD change 
the effect (and the outcome). It makes my head hurt (and doing 
algebra in an email box is almost as bad ;-). Can you give me a rule 
for where to put square roots in the order of ops?  Thank you!

Date: 10/09/2009 at 17:55:55
From: Doctor Rick
Subject: Re: When do you do a square root in the order of ops

Hi, Tracey.

I'll write the expression this way:

  14 + (24 - 12)^2 / 2 * 3^3 + (4 - 2^2) + sqrt(9)

The part you're most concerned about is the last term; it probably 
looked a bit more like this on the paper:
    ___
  \/ 9

One thing you need to notice is the bar (vinculum) over the 9. This 
may not be taught very often, but the vinculum is actually a remnant 
of an old alternate to parentheses. It is a grouping symbol, and as 
such, it goes along with parentheses in the order of operations. It's
attached to the radical sign that says to take the square root; 
therefore immediately after evaluating the expression under the 
vinculum, you can take its square root.

Thus, the first thing to do is to evaluate each quantity in 
parentheses, AND evaluate the root:

  14 + (24 - 12)^2 / 2 * 3^3 + (4 - 2^2) + sqrt(9)
       \_______/               \_______/   \_____/
  14 +     12   ^2 / 2 * 3^3 +     0     +    3
           \_____/       \_/
  14 +       144   / 2 *  27 +     0     +    3
             \_______/
  14 +           72    *  27 +     0     +    3
                 \_________/
  14 +               1944    +     0     +    3

Now, with only additions left, we add them up and get 1961. I get 
the same answer you got, though I evaluated the square root first. 
All that really matters here, as you observed, is that the square 
root is evaluated before the final addition.

The order of operations really defines a "partial ordering" of the 
operations in a given expression; certain operations in the 
expression must be done before certain others, but the ordering of 
other pairs of operations may not matter. In fact, properties such 
as the associative property open up even more possible orderings, 
for instance, additions don't really need to be done left to right.

If you can find any case where you think some other order of 
operations (in regard to square roots) would change the result, let 
me see it. Personally, I can't imagine doing anything else; someone 
less familiar with roots may see an issue that I don't see.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/