B. Number · Operations: Exponents, Factors, Fractions, Decimals, Integers, Percents and Proportions

The Order of Operations with Exponents

B2.1 B2.7 A1About 45 minutes

Learning goals

I can use the order of operations, including exponents, to find the value of an expression.
I can use brackets to change the order of operations.
I can use the properties of operations to calculate in my head.
I can check a calculation using the inverse operation.

Words to know

expression
Numbers and operation signs together, like 3 + 2³ × 4. It has a value, but no equal sign.

BEDMAS
A way to remember the order of operations: Brackets, Exponents, Division and Multiplication (left to right), Addition and Subtraction (left to right).

commutative property
You can add or multiply in any order: 4 × 7 = 7 × 4.

associative property
You can group numbers in any way when you add or multiply: (25 × 4) × 9 = 25 × (4 × 9).

distributive property
You can split a number to multiply: 7 × 98 = 7 × 100 − 7 × 2.

inverse operations
Operations that undo each other, like multiplying and dividing.

Remember from Grade 6

In Grade 6 you learned to work out expressions in this order: brackets first, then multiply and divide from left to right, then add and subtract from left to right. You skipped the E in BEDMAS. Now that you know exponents, the E joins the list.

Let's start with a puzzle

Amir and Lucie both work out 3 + 2³ × 4.

Amir: 3 + 2 = 5, then 5³ = 125, then 125 × 4 = 500.
Lucie: 2³ = 8, then 8 × 4 = 32, then 3 + 32 = 35.

Lucie is right. Exponents come before multiplying and adding, and Amir also added first. The order of operations makes sure everyone gets the same value.

Same expression, two orders. Doing the exponent first, then multiplying, then adding gives the correct value, 35.

Order	Step	Example: 50 − 2 × (1 + 3)²
1	Brackets	1 + 3 = 4, so 50 − 2 × 4²
2	Exponents	4² = 16, so 50 − 2 × 16
3	Division and Multiplication, left to right	2 × 16 = 32, so 50 − 32
4	Addition and Subtraction, left to right	50 − 32 = 18

BEDMAS: the value of 50 − 2 × (1 + 3)² is 18.

Partners work left to right

D and M are partners: do whichever comes first from the left. A and S are partners too. In 24 ÷ 4 × 2, divide first (24 ÷ 4 = 6), then multiply (6 × 2 = 12).

Worked example

Example 1: Brackets and exponents

Find the value of (3 + 2)² and of 3² + 2².

(3 + 2)²: brackets first. 3 + 2 = 5. Then 5² = 25.
3² + 2²: exponents first. 3² = 9 and 2² = 4. Then 9 + 4 = 13.

(3 + 2)² = 25 but 3² + 2² = 13. Brackets change the value.

Worked example

Example 2: A longer expression

Find the value of 36 ÷ 3² + 5 × (8 − 6).

Brackets: 8 − 6 = 2. Now it is 36 ÷ 3² + 5 × 2.
Exponents: 3² = 9. Now it is 36 ÷ 9 + 5 × 2.
Divide and multiply from left to right: 36 ÷ 9 = 4 and 5 × 2 = 10. Now it is 4 + 10.
Add: 4 + 10 = 14.

36 ÷ 3² + 5 × (8 − 6) = 14

Rewrite the shorter expression after every step. Each line is worth the same amount: 14.

Properties make mental math easier

The properties of operations let you rearrange a calculation into a friendlier one. They work for whole numbers, decimals and fractions.

Any order (commutative): 0.5 × 37 × 2 = 0.5 × 2 × 37 = 1 × 37 = 37.
Any grouping (associative): 25 × 36 = 25 × (4 × 9) = (25 × 4) × 9 = 100 × 9 = 900.
Splitting (distributive): 7 × 98 = 7 × (100 − 2) = 700 − 14 = 686.

Relationships between operations: multiplication and division undo each other, and so do addition and subtraction. Use them to check your work. Exponents are repeated multiplication, so you can check 2⁵ = 32 by dividing 32 ÷ 2 five times to get back to 1.

Worked example

Example 3: Use properties and check

Find 8 × 4.5 × 1.25 in your head, then check it.

Change the order: 8 × 1.25 × 4.5.
Group the friendly pair: 8 × 1.25 = 10. (1.25 + 1.25 = 2.5, doubled is 5, doubled again is 10.)
Then 10 × 4.5 = 45.
Check with the inverse operation: 45 ÷ 4.5 = 10 and 10 ÷ 1.25 = 8. ✓

8 × 4.5 × 1.25 = 45

Think like a mathematician

Before you start, read the whole expression. Circle the brackets, then mark the exponents. Rewrite the shorter expression after every step. If two people get different answers, compare your written steps to find where you split apart. This is how mathematicians check each other's reasoning.

Hands-on activity

Dice target game

You need: 4 dice (or number cards 1 to 6 made from paper), paper and a pencil. Play with a grown-up or sibling.

Roll one die twice to make a two-digit target (for example, 4 then 2 makes 42).
Roll 4 dice.
Each player uses all 4 numbers once to get as close to the target as possible. You may use +, −, ×, ÷, brackets and one exponent of 2 or 3 (squared or cubed).
Swap papers. Your partner checks your expression using BEDMAS.
The closest correct expression wins a point.

Example: target 42, dice 1, 3, 5 and 6. One way: 6 × (5 + 3 − 1) = 6 × 7 = 42. A try with an exponent: 6² + 5 + 3 − 1 = 43, which is only 1 away.

Math talk: "Where did brackets help you? Where did an exponent help you?"

Just for fun

Why did the exponent get to go before the multiplication sign?

It had a BEDMAS pass.

Your turn

2 + 3² = ?

(2 + 3)² = ?

4 × 2³ = ?

20 − 4² ÷ 2 = ?

6² − 5 × 6 = ?

3 × (10 − 2³) = ?

Use a property to find 5 × 99 in your head.

More practice: warm-up

10 − 3² = ?

2³ + 1 = ?

(1 + 1)³ = ?

5² × 2 = ?

100 ÷ 5² = ?

More practice: on your own

2 × 3² + 4 × (5 − 3)³ = ?

(12 − 2²) ÷ 2 + 1 = ?

0.3² = ?

(1/2)³ = ?

Put brackets in 2 + 3 × 4² so the value is 80.

Challenge

Sam says 2 × 3² = 36. What mistake did Sam make? What is the correct value?

A community garden has one square bed with sides of 6 m and three square beds with sides of 2 m. Write one expression for the total area and find its value.

Use four 4s and any operations (and brackets) to make 9 and to make 7.

Check your understanding

Quiz

What is 5 + 2³?

What is (4 + 1)²?

In 18 − 2 × 3², what do you do first?

Which is the easiest way to find 3 × 98 in your head?

What is 48 ÷ 2² × 3?

What is 0.2²?

For parents and teachers

Big idea: The Ontario curriculum (B2.1) asks Grade 7 students to use the properties and order of operations, and the relationships between operations, to solve problems involving whole numbers, decimals, fractions, ratios, rates and percents, including multi-step problems. With exponents now part of Grade 7 (B2.7), the full BEDMAS order applies.

Common difficulties:

Treating BEDMAS as six separate steps and always multiplying before dividing. Remind your child that D/M and A/S are partners done left to right.
Squaring before doing the brackets, or working out 2 × 3² as 6².
Squaring decimals: 0.3² = 0.09, not 0.9.

How to help: Play the dice target game and ask your child to rewrite the expression after each step. Praise clear written steps as much as the right answer. This lesson builds A1 skills: choosing efficient strategies, persevering, and explaining reasoning to others.