The Order of Operations

Parenthesis, exponents, multiply, divide, add and subtract are the order of operations. The order of operations is the order in which numeric mathematical operations should be solved. When there is a function in the equation, it must be solved before the PEMDAS operations can be solved. The standard was put in place for the stability of results in mathematical operations. I will include four examples to walk through each step in solving equations that either use all or part of these methods to arrive at the correct standard results.

Before I begin with the first example, it is my preference to list and number the methods in sequential order in the beginning of this paper for a simple, logical reference.

  1. Parenthesis
  2. Exponents
  3. Multiply & Divide
  4. Add & Subtract

First example:

One may naturally want complete an equation like this 3+9 x 8 – 6 ^ 2 from left to right: is 3+9= 12×8= 96-6= 90^2= 8100. Another person may start with the addition and subtraction solutions first 3+9= 12   8-6=2    12 x 2 ^ 2= 12 x 4= 48. As you can see, the result of the very same equation was very different by simply mixing up the order each part was solved. This is a prime example of showing why we need a standard mathematical order of operations.

3+9 x 8 – 6 ^ 2 has no parenthesis, so the next order of operations is to solve the exponents.

3+9 x 8 – 36 next orders are multiply and divide:

3+72 – 36 and last orders are to add and subtract what is left:

3+72= 75-36= 39 And because there is a standard order for this equation, no matter what order it is in will return the same result. If it does not, then somewhere you have an error. This is the same equation given in a different order, but solved in the correct order of operations. Once again the original equation was 3+9 x 8 – 6 ^ 2. The new order is as follows: -6^2 + 9 x 8 + 3 = exponent first -36+72+3=multiply second -36+75= 39 and finally add and subtract. These examples prove the value of a standard order of operations.

Second example:

The next equation is another good example with each of the numeric math operations represented. 16 + (4^2 – 2 x 3) ^ 2 – 9 = first the exponents then multiplication within the parenthesis: 16 + (16 – 6) ^2 – 9= then solve the rest of what is in the parenthesis and solve the exponential value of 2 for what is in the parenthesis: 16 + 100 – 9 = then solve with addition and subtraction: 107.

Third and Fourth (Final) examples:

The equation 16-8+3-2+4 can be solved from left to right: 16-8= 8+3= 11-2= 9+4= 13. 16-8+3-2+4= let’s solve the subtraction first: 8+1+4= 13. The order in which you solve this equation matters not. Multiplication and division don’t care either. This is the problem 24 / 6 x 2 x 2 / 4 solved from left to right = 24 / 6= 4 x 2= 8 x 2= 16 / 4 = 4.  If you decide to solve all of the division first, you would have 4 x 2 x ½ = 8 x ½ = 4.  So, it is important to remember the order of operations. But multiply and divide, or add and subtract are both pairs that don’t care who goes first when it’s just the two of them.

Reference:

Math Forum (1994-2008).  Math Forum: Ask Dr. Math FAQ: Order of
Operations
. Retrieved January 26, 2009 from Drexel University,
Web site:

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