Order Precedence (Order of Operations)

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:

1. 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.
2. Exponents get evaluated after grouped expressions.
3. Multiplication and division follow, evaluated left to right.
4. 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 Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction.

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.