# 1.07 Simplifying numerical expressions involving integers

Lesson

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:

Order of operations
1. 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.
2. Evaluate all exponential expressions, such as squares and cubes.
3. Multiply and/or divide in order from left to right.
4. 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$1009×6+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)÷​210÷​(2)

##### Question 6

Evaluate $9^2+6$92+6

### Outcomes

#### 7.NS.1

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

#### 7.NS.1.d

Apply properties of operations as strategies to add and subtract rational numbers.

#### 7.NS.2

Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

#### 7.NS.2.c

Apply properties of operations as strategies to multiply and divide rational numbers.