**Order Precedence** tells us the order in which we need to evaluate mathematical equations. It is also widely known as the order of operations.

The Order of Operations is as follows:

- First, evaluate any grouped expressions, such as expressions in parentheses, over and under vinculums (horizontal division lines), and radical expressions (square roots). Start with the innermost grouping.
- Exponents get evaluated after grouped expressions.
- Multiplication and division follow, evaluated left to right.
- Addition and subtraction are evaluated last, left to right.

A commonly taught acronym is PEMDAS, or “Please Excuse My Dear Aunt Sally.” This stands for **P**arenthesis, **E**xponents, **M**ultiplication, **D**ivision, **A**ddition, **S**ubtraction.

We will now look at three examples showing proper order precedence.

#### Example One

Take the following equation: \(y=5+2*4\)

If it helps, you can add parenthesis to equations so you remember the proper order to evaluate. \(y=5+(2*4)\)

First, we evaluate \(2*4\), because multiplication comes before addition. So we now have \(y=5+8\).

Next, we evaluate \(5+8\), which is 13. So our final answer is \(y=13\).

Note that if we did this **wrong**, and evaluated \(5+2\) before \(2*4\), we would get \(y=7*4\) which is equal to 28 and is incorrect.

#### Example Two

Now let’s take a slightly more complex example: \(y=3+4*\sqrt{12+2^2}\)

We need to evaluate the radical first, because it’s grouped. Since exponents come before addition, we must evaluate \(2^2\) before \(12+2\).

So we evaluate \(2^2=4\) and return this value to the equation: \(y=3+4*\sqrt{12+4}=3+4*\sqrt{16}\)

The square root of 16 is 4, so \(y=3+4*4\). Finally, we need to multiply before we add, so \(y=3+16=19\).

#### Example Three

Finally, let’s take a look at an equation with a lot of things going on:

\(y=\frac{3+5*(4-1)}{4+\sqrt{8*2}}\)

Start by evaluating the grouped expressions. So \((4-1)\) becomes 3, and \(\sqrt{8*2}\) becomes \(\sqrt{16}\) or 4.

Plug these values back into the original equation:

\(y=\frac{3+5*(3)}{4+4}\)

We multiply \(5*3=15\), then we evaluate the top and bottom of the fraction separately: \(y=\frac{3+15}{8}\).

So our final answer is: \(y=\frac{18}{8}\), which simplifies to \(\frac{9}{4}\) or 2.25.