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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. 

Be especially careful with problems like the following.

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

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

  $3+4\left(5+6\right)\ne 7\left(5+6\right)$ 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.

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

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

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