Order of Operations

Date: 01/25/98 at 04:26:11
From: ANDREW HATFIELD
Subject: MATH

Evaluate the expression:

    2  3
30-5 +4

Date: 01/28/98 at 12:44:21
From: Doctor Joe
Subject: Re: MATH

Dear Andrew,

First of all, let's realise that there are three major operations 
involved in this question:

(1)  subtraction
(2)  addition
(3)  multiplication.

Though it does not appear in your question, there is another 
operation, called division.

I assume that you know how to do each operation on its own; for 
example, you already know how to compute:

(a)  23 + 34 = 57
(b)  40 - 11 = 29
(c)  23 x 4  = 92

You can see that there may be some trouble if the operations are 
written in a linear sequence; for example:

   3 - 4 x 5

One does not know which operation to perform first.

The solution to the problem is this: by convention, we compute the 
sequence using these rules:

(1) Perform the operation first if the terms involved are bracketed. 
    For instance:

    2 - (3 + 5)

    We do the (3 + 5) bit first!  So, the next step is:

    2 - (3 + 5) = 2 - 8

    Okay so far?

(2) After evaluating bracketed terms, perform multiplication or  
    division in that order. For example:

    3 + 4 x 5 / 2    (where / means divided by)

    So, multiply 4 and 5 together first. That's 20.

    Then we divide it by 2 to get 10.

    Next, we add 3 to 20. We obtain 23 as the final answer.

(3) Last of all perform addition or subtraction in that order; for 
    example:

    3 - 4 + 5

    So we subtract 4 from 3, to obtain -1.  Then we add -1 to 5 to 
    obtain 4 as the final answer.

Let's try practising these rules in the following problems:

(1)  20 + 34 - 4 x 5

     Answer:  20 + 34 - 4 x 5 = 20 + 34 - 20 = 54 - 20 = 34.

(2)  30 - 5 + 4 x 2

     Answer:  30 - 5 + 4 x 2 = 30 - 5 + 8 = 25 + 8 = 33.


Finally, let's discuss some notation found in your question:
           2  
What does 5  (also written as 5^2, using my notation) mean?

Well, it simply means 5 x 5, i.e. 5 multiplied by itself, and together 
you have a product of two 5's.

In general, n^m means n x n x n x ... x n (altogether, m copies of 
n's).

Here's another example:  4^6 = 4 x 4 x 4 x 4 x 4 x 4.

There are 6 copies of 4.


Let's do one more problem:

Question:  2^4 - 5^2 + 100

Answer:  2^4 - 5^2 + 100 = 2 x 2 x 2 x 2 - 5 x 5 + 100
                         =     4 x 2 x 2 - 25    + 100
                         =         8 x 2 - 25    + 100
                         =            16 - 25    + 100
                         =                 -9    + 100
                         = 91

Working out your problem:

    2  3
30-5 +4   = 30 - 5 x 5 + 4 x 4 x 4

You complete the rest of the problem.

Feel free to write if you have further queries.

-Doctor Joe,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/