# Order of Operations

Math has a defined and precise order of operations that must be followed for problems to come out correctly. Calculators are programmed with the order of operations. Students have to learn them!

Most students learn the acronym PEMDAS to remember the order of operations.  PEMDAS stands for:

1st do Parentheses (do operations in parentheses first -- following PEMDAS inside the parentheses!  Do nested parentheses from the most nested (deepest inside the parentheses) to the least nested)

2nd do Exponenents (simplify any exponents)

3rd do Multiplication and Division (do all multiplication and division problems, from left to right)

4th do Addition and Subtraction (do all addition and subtraction problems, from left to right  -- but that's all that should be left at this point!).

Because multiplication and division are performed together (which comes first is determined by left to right placement) and addition and subtraction are performed together (again, specific order determined by left to right placement), we prefer to write PEMDAS as:

P
E
MD
AS

With one step problems, you don't have to worry about order of operations, but as soon as there is more than one step, order of operations matters.

Example:

$4+6\times7=$

There are no parentheses.
There are no exponents.
Do the multiplication, then do the addition:

\eqalign{4+6\times7&=\\4+\underbrace{6\times7}&=\\4+42&=48}

A student who did not know order of operations would probably have done $4+6$ first and gotten $70$, but the correct answer is $48$.  If you don't believe it, try it on a calculator!

Simple problems, like the example above, are sometimes the easiest ones to miss.  Students often don't even think they need order of operations for such short problems.  But, you have to be very careful working through more complicated problems, just to make sure you catch everything. However, the process, of following PEMDAS remains the same:

Example:

$41-(8^2\div4)+3=$

We have to do the operations inside the parentheses first.
Inside the parentheses there are no parentheses.
Inside the parentheses there is an exponent.
Then there is division.

\eqalign{41-&(\underbrace{8^2}\div4)&&+3=\\41-&\underbrace{(64\div4)}&&+3=\\41-&\quad16&&+3=}

Now that the parentheses are cleared out:
We have no more parentheses.
We have no exponents.
We have no multiplication or division.
We do addition and subtraction from left to right.

\eqalign{41-16&+3=\\\underbrace{41-16}&+3=\\25\quad&+3=28}

The trickiest part of order of operations is that it ALWAYS applies. It applies in algebra, geometry, even little kid math. So, whenever you see a multi-step equation, go through the steps of PEMDAS to make sure that you're doing your operations in the correct order.

Note: Use parentheses to CHANGE the order in which a problem is done! Look back to the first example. What if you wanted to do the addition first?

$4+6\times7=48 \qquad \text{, but} \qquad (4+6)\times7=70$

According to the orders of operations, in the first problem you have to do the multiplication first.  But, in the second problem, because $(4+6)$ is in parentheses, you do that first, changing the problem entirely. Overall, you don't need parentheses when you want the person doing the math to follow order of operations, but if you want to change the order of operations for a problem, use parentheses!

• ## Order of Operations

1. $30 - (2^2 +2) \cdot 3$

2. $8^2 \div (3+1)^2 + 2$

3. $(5-3)^5 +3^3$

4. $4^3 \div (10-6)^2 + 9$

5. $9^2 \cdot (10-6)^2 + 9$

6. $42 - (7^2 - 40) - 3$

7. $32 + (3^3 - 18) + 4$

8. $(8-6)^4 + 7 \cdot 3^2$

9. $8 + (32-23)^2 \div 3$

10. $42 - (3^2 + 2) \cdot 3$