Let's Review
The order of operations tells you how to simplify an expression that contains more than one operation. It is shown below.
| P | Parentheses |
|---|---|
| E | Exponents |
| M D |
Multiplication and Division from left to right, in the order they appear. |
| A S |
Addition and Subtraction from left to right, in the order they appear. |
Tip
When using the order of operations, it is tempting to skip writing down steps or to do certain operations in your head. When simplifying these types of problems, it is important to write down all your steps so that you know you are simplifying in the correct order and also so that it is easier to find any errors you did make.
You should also note that the order of operations can be applied to all number sets. When you simplify these expressions, expect to see whole numbers, fractions, decimals, and integers.
For Example
Simplify \( 2^{4} - \left( 16 - 10 \right) + 4 \cdot 5 \).
Parentheses |
Simplify the expression inside the parentheses. \( 2^{4} - \require{color}\colorbox{yellow}{$ \left( 16 - 10 \right) $} + 4 \cdot 5 \) \( 2^{4} - \require{color}\colorbox{yellow}{$ 6 $} + 4 \cdot 5 \) |
Exponents. |
Simplify the number written in exponential notation. \( \require{color}\colorbox{yellow}{$ 2^{4} $} - 6 + 4 \cdot 5 \) \( \require{color}\colorbox{yellow}{$ 16 $} - 6 + 4 \cdot 5 \) |
Multiplication and division from left to right. |
This expression contains only multiplication. \( 16 - 6 + \require{color}\colorbox{yellow}{$ 4 \cdot 5 $} \) \( 16 - 6 + \require{color}\colorbox{yellow}{$ 20 $} \) |
Addition and subtraction from left to right. |
Perform the subtraction and then the addition. \( \require{color}\colorbox{yellow}{$ 16 - 6 $} + 20 \) \( \require{color}\colorbox{yellow}{$ 10 $} \require{color}\colorbox{lime}{$ + 20 $} \) \( \require{color}\colorbox{lime}{$ 30 $} \) |
Let's Practice
Practice performing the order of operations by completing the activity below. Simplify each expression using the order of operations. Then check your answer.
Simplify \( 7 \div 7 + 3 - 2 \cdot 5 \).
\( - 6 \)
If you need help arriving at this answer, click the solution button.
There are no parentheses or exponents. Perform the multiplication and division from left to right. |
\( \require{color}\colorbox{lime}{$ 7 \div 7 $} + 3 - 2 \cdot 5 \) \( \require{color}\colorbox{lime}{$ 1 $} + 3 - \require{color}\colorbox{yellow}{$ 2 \cdot 5 $} \) \( \require{color}\colorbox{lime}{$ 1 $} + 3 - \require{color}\colorbox{yellow}{$ 10 $} \) |
Complete the addition and subtraction from left to right. Use what you know about adding and subtracting integers. When the signs are unlike, subtract the smaller absolute value from the larger absolute value. The sum will have the same sign as the number with the larger absolute value. |
\( \require{color}\colorbox{yellow}{$ 1 + 3 $} - 10 \) \( \require{color}\colorbox{yellow}{$ 4 $} \require{color}\colorbox{lime}{$ - 10 $} \) \( \require{color}\colorbox{yellow}{$ 4 $} \require{color}\colorbox{lime}{$ + \left( - 10 \right) $} \) \( \require{color}\colorbox{lime}{$ - 6 $} \) |
Simplify \( 6 + 21 \div 3 \cdot 4 + 3^{3} \).
\( 61 \)
If you need help arriving at this answer, click the solution button.
There are no parentheses, so start by simplifying the exponent. |
\( 6 + 21 \div 3 \cdot 4 + \require{color}\colorbox{yellow}{$ 3^{3} $} \) \( 6 + 21 \div 3 \cdot 4 + \require{color}\colorbox{yellow}{$ 27 $} \) |
Multiplication and division from left to right, in the order they appear. |
\( 6 + \require{color}\colorbox{yellow}{$ 21 \div 3 $} \cdot 4 + 27 \) \( 6 + \require{color}\colorbox{yellow}{$ 7 $} \require{color}\colorbox{lime}{$ \cdot\ 4 $} + 27 \) \( 6 + \require{color}\colorbox{lime}{$ 28 $} + 27 \) |
Addition from left to right, in the order they appear. |
\( 6 + 28 + 27 \) \( 61 \) |
Simplify \( 3^{3} - 8 \div 4 \cdot 2 + 7 \).
30
If you need help arriving at this answer, click the solution button.
Exponents. |
\( \require{color}\colorbox{violet}{$ 3^{3} $} - 8 \div 4 \cdot 2 + 7 \) \( \require{color}\colorbox{violet}{$ 27 $} - 8 \div 4 \cdot 2 + 7 \) |
Multiplication and division from left to right, in the order they appear. |
\( 27 - \require{color}\colorbox{yellow}{$ 8 \div 4 $} \cdot 2 + 7 \) \( 27 - \require{color}\colorbox{yellow}{$ 2 $} \require{color}\colorbox{lime}{$ \cdot 2 $} + 7 \) \( 27 - \require{color}\colorbox{lime}{$ 4 $} + 7 \) |
Addition and subtraction from left to right, in the order they appear. |
\( 27 - 4 + 7 \) \( 23 + 7 \) \( 30 \) |
Simplify \( {1.5}^{2} \div 0.5 \cdot 5 - (10 + 5) \)
7.5
If you need help arriving at this answer, click the solution button.
Parentheses |
\( {1.5}^{2} \div 0.5 \cdot 5 - \left( \require{color}\colorbox{yellow}{$ 10 + 5 $} \right) \) \( {1.5}^{2} \div 0.5 \cdot 5 - \require{color}\colorbox{yellow}{$ 15 $} \) |
Exponents |
\( \require{color}\colorbox{yellow}{$ {1.5}^{2} $} \div 0.5 \cdot 5 - 15 \) \( \require{color}\colorbox{yellow}{$ 2.25 $} \div 0.5 \cdot 5 - 15 \) |
Multiplication and division from left to right, in they order they appear. |
\( \require{color}\colorbox{lime}{$ 2.25 \div 0.5 $} \cdot 5 - 15 \) \( \require{color}\colorbox{lime}{$ 4.5 $} \require{color}\colorbox{yellow}{$ \cdot 5 $} - 15 \) \( \require{color}\colorbox{yellow}{$ 22.5 $} - 15 \) |
Subtraction. |
\( 22.5 - 15 \) \( 7.5 \) |
Humza works for a large zoo, ensuring the zoo always has enough food for the animals. Every two weeks, she needs to order 780 pounds of frozen fruit. This morning Humza was getting ready to place the order for the next two weeks when one of the zookeepers told her that because of a temporary diet change for monkeys, they still have 125 pounds of frozen fruit in the kitchen. But starting next week, the monkeys will be back to their regular diet.
If frozen fruit costs $2.04 per pound, how much will the zoo spend to have 780 pounds of frozen fruit on hand for the next two weeks?
\( \$ 1,336.20 \)
If you need help arriving at this answer, click the solution button.
Parentheses |
\( 2.04 \cdot \left( \require{color}\colorbox{yellow}{$ 780 - 125 $} \right) \) \( 2.04 \cdot \require{color}\colorbox{yellow}{$ 655 $} \) |
Multiplication |
\( 2.04 \cdot 655 \) \( 1336.20 \) |
