Order of Operations

In mathematics the order of operations is a collection of rules that define which procedures to perform first, so that when we evaluate a given mathematical expression, we will all come up with the same answer.

1 st : Do any calculations inside p arentheses or other grouping symbols, starting with the innermost and working out.

2 nd : Simplify any e xponential expressions.

3 rd : Work all m ultiplications and d ivisions, from left to right, as they appear.

4 th : Work all a dditions and s ubtractions, from left to right, as they appear.

So you don't get confused, remember PEMDAS which stands for Parentheses, Exponents, Multiplication-Division, Addition-Subtraction.

In California, we say P owerful E arthquakes M ay D eliver A fter- S hocks.

Example :

Simplify $3+2×{\left(5-7\right)}^{2}$ .

Do the operation in parenthesis first.

$=3+2×{\left(-2\right)}^{2}$

Then evaluate the exponent. Since the negative sign is inside the parenthesis, this means $\left(-2\right)×\left(-2\right)$ .

$=3+2×4$

$=3+8$

$=11$

Be especially careful with problems like the following.

${\left(3×4\right)}^{2}={12}^{2}=144$ because parentheses come before exponents, BUT $3×{4}^{2}=48$ because exponents come before multiplication.

${\left(-4\right)}^{2}=\left(-4\right)\left(-4\right)=16$ BUT $-4×4=-16$

$3+4\left(5+6\right)\ne 7\left(5+6\right)$ because parenthesis is the operation to start with.

So, $3+4\left(5+6\right)=3+4\left(11\right)=3+44=47$ .

Also, be careful with fractions. The fraction bar acts like a grouping symbol, so simplify the numerator and denominator first.

$\begin{array}{l}\frac{5\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4}{8\text{\hspace{0.17em}}-\text{\hspace{0.17em}}5}=\frac{9}{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=3\end{array}$

You can think of it this way: if you rewrote the fraction on one line, using the division symbol, you would need parentheses.

$\begin{array}{l}\frac{5\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4}{8\text{\hspace{0.17em}}-\text{\hspace{0.17em}}5}=\left(5+4\right)÷\left(8-5\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=9÷3\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=3\end{array}$