1.07 Simplifying numerical expressions involving integers
In life, the order in which we do things is important. For example, we put on socks then shoes, rather than shoes and then socks.
The same goes for evaluating expressions in math with more than one operation. There are a number of rules which need to be followed in order to solve these problems correctly. The order goes:
- Complete all operations within grouping symbols such as parentheses (...) or absolute values |...|. If there are grouping symbols within other grouping symbols, do the innermost operation first.
- Evaluate all exponential expressions, such as squares and cubes.
- Multiply and/or divide in order from left to right.
- Add or subtract in order from left to right.
Worked examples
question 1
Evaluate: $5\times\left(6+6\right)$5×(6+6)
Think: Remember the order of operations. Firstly, we'll evaluate what's inside the parentheses, then we'll evaluate the multiplication.
Do:
| $5\times\left(6+6\right)$5×(6+6) | $=$= | $5\times12$5×12 |
Add within the grouping symbol |
| $=$= | $60$60 |
Multiply |
Reflect: How would this be different if there were no parentheses?
question 2
Evaluate: $100-9\times6+18\div6$100−9×6+18÷6
Think: There are no grouping symbols in this question, so firstly we'll evaluate the multiplication and division (going from left to right), then we will evaluate the addition and subtraction (going from left to right).
Do:
| $100-9\times6+18\div6$100−9×6+18÷6 | $=$= | $100-54+3$100−54+3 |
Evaluating $9\times6=54$9×6=54 and $18\div6=3$18÷6=3 |
| $=$= | $46+3$46+3 |
Evaluating $100-54$100−54 |
|
| $=$= | $49$49 |
Simplifying |
The two examples above used only whole numbers. However, we can work through order of operations problems with any type of real number such as integers, fractions or decimals. Let's look at some involving integers.
Question 3
Evaluate: $\left(48\div\left(-12\right)-35\div5\right)\times3^2$(48÷(−12)−35÷5)×32
Think: We need to simplify the problem by using our order of operation rules.
- Firstly, we evaluate what's inside the parentheses - division first and then subtraction.
- Then, evaluate the exponents.
- Then, we evaluate any other multiplication or division, working from left to right.
Do:
| $\left(48\div\left(-12\right)-35\div5\right)\times3^2$(48÷(−12)−35÷5)×32 | $=$= | $\left(-4-7\right)\times3^2$(−4−7)×32 |
Within the parentheses we start with $48\div\left(-12\right)=-4$48÷(−12)=−4 and $35\div5=7$35÷5=7 |
| $=$= | $-11\times3^2$−11×32 |
Now within the parentheses we can evaluate $-4-7$−4−7 |
|
| $=$= | $-11\times9$−11×9 |
Next we can evaluate the exponent $3^2=9$32=9 |
|
| $=$= | $-99$−99 |
Perform the multiplication |
Reflect: How might a number line help us to simplify this question?
Practice questions
Question 4
Evaluate $-18+21\div7$−18+21÷7
Question 5
Evaluate $\left(-14\right)\div2-10\div\left(-2\right)$(−14)÷2−10÷(−2)
Question 6
Evaluate $9^2+6$92+6