Order of Operations: Parentheses as Packages

Date: 10/25/2001 at 16:40:49
From: Kristina
Subject: Order of Operations

I need help figuring out the steps to this expression:

   5 - { -4 [2^4 - 11 ( -9 - 1 )] } =

Thanks!

Date: 10/25/2001 at 17:11:58
From: Doctor Peterson
Subject: Re: Order of Operations

Hi, Kristina.

The main feature of "order of operations" involved here is the 
parentheses. You can think of them as packages; you have to open the 
package in order to use it, which you do by finding the value of the 
expression inside. You have to treat the whole package as one number.

So if you start at the left and try to evaluate this expression, you 
will say "I start with 5; then I subtract - ah, I'll have to put that 
on hold while I figure out what's inside the braces - Let's start over 
now, I have -4 and I have to multiply it by - oops, we're on hold 
again until I find out what's inside the brackets ...".

That sounds confusing, doesn't it? You can save all that confusion by 
evaluating the whole thing from the inside out. You know you will have 
to find the value of the innermost "package" before you can do 
anything else, so you can just do that first. I'll demonstrate:

    5 - { -4 [2^4 - 11 ( -9 - 1 )] }
                        \______/
    5 - { -4 [2^4 - 11 *   -10   ] }

What I just did was to find the innermost parenthetical expression, 
evaluate it as -10, and put that in place of the expression. I also 
put in a multiplication sign, since the 11 next to the parentheses 
means to multiply. Now you can do the same thing with the bracketed 
expression [], and then with the braces {}.

When you evaluate the next part, remember the second rule of order: 
you will first multiply 11 by -10, and then subtract that from 2^4.

For more on this concept, see the Dr. Math FAQ:

   Order of Operations
   http://mathforum.org/dr.math/faq/faq.order.operations.html   

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