In this explainer, we will learn how to simplify and evaluate numerical and algebraic expressions using the order of operations.
In mathematics, we will often be working with numerical values. These expressions become more complicated the more operations that are needed in the expression. So, to help reduce this complexity, we define an order of operations to tell us the order in which we should evaluate the operations in any expression.
Before we define the order of operations, let’s first see a specific example of why it might be useful to define the order of operations. Consider the expression . We first recall that we can represent the order of operations using parentheses. In particular, the operations inside the parentheses must be carried out first.
If we evaluate the multiplication first, we get
However, if we evaluate the sum first, we get
These give different values, so the order in which we evaluate the operations will change the outcome of the expression.
We can represent the order by always using parentheses. However, as mentioned earlier, the more operations in an expression, the more complicated this would become. Instead, we define an order in which we carry out the operations if no parentheses are present to help us simplify these expressions.
Definition: Order of Operations
In any expression, we carry out the operations in the following order:
We can remember this order with the acronym PEMDAS.
Thus, if we go back to our previous expression , we note that multiplication is evaluated before addition. So,
If we wanted to write this expression so that the addition is carried out first, we would need to include parentheses:
It is worth noting that it is also common to write the order of operations as follows:
- Multiplication and division
- Addition and subtraction
These are equivalent ways of thinking about the order of operations since multiplication and division on one hand and addition and subtraction on the other hand are similar operations, as we will explain in our first example.
Example 1: Understanding the Order of Operations
The operations are in the wrong order. Use the letters to list them in the correct order.
We can recall the order of operations by using the acronym PEMDAS. The order of operations is as follows:
We note that parentheses are the same as groupings. It is also worth noting that addition and subtraction are very similar operations and can be evaluated in any order since , so any subtraction is also an addition. Similarly, multiplication and division are very similar operations since . Thus, any division by a nonzero value is also a multiplication. Therefore, we can also evaluate these operations in either order. These properties both follow from the commutativity and associativity of the addition and multiplication operations.
Hence, the order of the given operations is groupings, exponents, multiplication and division, and addition and subtraction. This is the order C, A, B, D.
In our next example, we will use the order of operations to evaluate a given numerical expression.
Example 2: Using Order of Operations to Evaluate Numerical Expressions Involving Exponents
We begin by recalling that the order of operations tells us the order in which we should evaluate operations in an expression. We can easily recall this order by using the acronym PEMDAS. The order of operations is as follows:
Thus, we will start by evaluating the expressions inside the parentheses. We have and . Substituting these values into the expression gives
Next, we need to evaluate the exponents. We can do this by using repeated multiplication. We have . Substituting this into the expression yields
We now see that there are no multiplication or division operations. So, we move on to addition. It is worth noting we can evaluate addition and subtraction in either order or at the same time.
Before we continue to the next example, there is something worth noting about fraction notation. When fraction notation is used, we want to evaluate the numerator and denominator before the division.
For example, to evaluate , we need to evaluate the numerator first, even though the operation is addition. One way of remembering this is to think of fractions as having parentheses:
Let’s see now another example of using the order of operations to evaluate a given numerical expression that includes fraction notation.
Example 3: Using Order of Operations to Evaluate Numerical Expressions Involving Exponents
Fill in the blank: .
We first recall that when using fractional notation, we need to evaluate the numerators and denominators before the division. We can then recall that the order of operations tells us to evaluate the operations in the following order:
We start with the expression in the parentheses. We need to then evaluate the numerators and denominators first. Evaluating the exponents gives and . Substituting these into the expression yields
Next, we evaluate the multiplication in the denominator. We have , so
Now, we can evaluate each numerator and denominator. This yields
Since we evaluated all of the numerators and denominators, we can evaluate the divisions for these fractions. We note that 10 and 16 share a factor of 2, so
We now need to add the fractions in the parentheses. We can do this by writing both to have a denominator of 24 as follows:
Finally, since 31 and 24 share no common factors, we evaluate the subtraction as follows:
In our next example, we will evaluate an algebraic expression by using the order of operations.
Example 4: Using the Order of Operations to Evaluate an Algebraic Expression at Given Values
Given that and , find the value of .
We first substitute the values of and into the expression to get
We can simplify this expression by noting that adding a negative sign is the same as subtraction, as shown:
We can now follow the order of operations; we can recall this order by using the acronym PEMDAS. The P stands for parentheses, so we want to start with the expressions inside the parentheses.
This is the expression . We can evaluate this subtraction by rewriting both fractions to have the same denominator. Since 4 and 5 share no common factors greater than 1, their lowest common multiple is their product . Thus, we rewrite both fractions to have a denominator of 20 and evaluate as follows:
We can substitute this value into the expression to get
Now, we can evaluate the exponent by squaring the numerator and denominator. We get
Let’s see another example of evaluating an algebraic expression by using the order of operations.
Example 5: Using the Order of Operations to Evaluate an Algebraic Expression at Given Values
Given that , , and , find the value of .
We could start by substituting the values of , , and into the expression; however, this would give us an expression with nestled fractions. It would be easy to make a mistake with so much notation in a single expression.
Instead, let’s directly apply the order of operations to the given algebraic expression. We first note that in fraction notation, we need to evaluate the numerator and denominator before the division. So, we should evaluate and separately. Since these are single operations, we can evaluate these directly.
We note that 2 and 3 have no common factors greater than 1, so their product is their lowest common multiple: . Thus, we rewrite both fractions to have a denominator of 6 and evaluate as follows:
We can substitute these values into the algebraic expression, where we will include parentheses in the numerator and denominator to show that these are evaluated first.
Although not necessary, we can now rewrite the division of these two fractions using the division symbol to help us see exactly what we need to calculate, as follows:
To divide two fractions, we multiply by the reciprocal of the divisor. This gives
Finally, we note that 25 and 24 share no common factors greater than 1, so we cannot simplify this fraction any further.
In our final example, we will determine the area of a trapezoid using a given formula and the order of operations.
Example 6: Solving a Word Problem Using the Order of Operations
The area of a trapezoid is . Find when , , and . Give your answer approximated to one decimal place.
We can start by adding the values of , , and onto the diagram.
We can substitute the values for these lengths into the formula for the area to get where we add in a multiplication symbol and parentheses around to help keep track of the original parts of the formula.
We now recall that the order of operations starts with the expressions inside the parentheses. This means we should start with . We have
We can substitute this into the equation to get
We now have an expression with only multiplication, and we recall that we can evaluate the multiplication in any order since it is an associative and commutative operation. Thus, we can just evaluate these products to get
Let’s finish by recapping some of the important points from this explainer.
- In any expression, we carry out the operations in the following order:
- We can remember the order of operations with the acronym PEMDAS.