Associated Topics || Dr. Math Home || Search Dr. Math

### 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/

Associated Topics:
Elementary Division
Elementary Multiplication
Elementary Subtraction

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search