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

Exponents: Repeated Multiplication

B2.7 A1About 40 minutes
Learning goals
  • I can write repeated multiplication using exponents, like 2 × 2 × 2 × 2 × 2 = 2⁵.
  • I can name the base and the exponent of a power and find its value.
  • I can use squares, cubes and powers of 10 in real situations.
  • I can write a prime factorization using exponents.

Words to know

power
A number written with a base and an exponent, like 2⁵. It is a short way to write repeated multiplication.
base
The number being multiplied over and over. In 2⁵ the base is 2.
exponent
The small raised number. It tells how many times the base is used as a factor. In 2⁵ the exponent is 5.
exponential form
A number written as a power, like 3⁴ instead of 3 × 3 × 3 × 3.
squared
Raised to the exponent 2. 5² is read "5 squared" and means 5 × 5.
cubed
Raised to the exponent 3. 4³ is read "4 cubed" and means 4 × 4 × 4.

Remember from Grade 6

In Grade 6 you wrote numbers as a product of prime factors, like 56 = 2 × 2 × 2 × 7. You also learned the order of operations and were told that the E in BEDMAS stands for exponents, which would come later. Later is now! Exponents give us a short way to write the same factor many times.

Let's start with a sheet of paper

Fold a sheet of paper in half. Now it has 2 layers. Fold it in half again: 4 layers. Again: 8 layers. Each fold doubles the layers.

After 5 folds you have 2 × 2 × 2 × 2 × 2 = 32 layers. Writing all those 2s is slow. Mathematicians write it as 2⁵. The small 5 says "use 2 as a factor 5 times".

1 fold2 layers= 2¹2 folds4 layers= 2²3 folds8 layers= 2³…5 folds32 layers= 2⁵
Each fold doubles the layers, so after n folds there are 2 × 2 × … × 2 = 2ⁿ layers. Five folds give 2⁵ = 32.

Base, exponent and power

In 2⁵ = 32:

  • 2 is the base: the factor that repeats.
  • 5 is the exponent: how many times the base is used as a factor.
  • 2⁵ is a power. Its value is 32.

We read 2⁵ as "2 to the power of 5" or "2 to the fifth".

Squared and cubed. An exponent of 2 is called squared, and an exponent of 3 is called cubed. That is because a square with sides of 4 units has an area of 4 × 4 = 4² = 16 square units, and a cube with edges of 4 units has a volume of 4 × 4 × 4 = 4³ = 64 cubic units.

Powers of 10. 10² = 100, 10³ = 1 000, 10⁴ = 10 000, 10⁶ = 1 000 000. The exponent tells you how many zeros follow the 1. That is why powers of 10 are used to describe place value.

3² = 3 × 3 = 94² = 4 × 4 = 16
A 3-by-3 square has 3 × 3 = 9 small squares, so 3² = 9. A 4-by-4 square has 4 × 4 = 16 small squares, so 4² = 16. That is why an exponent of 2 is called squared.
Repeated multiplicationExponential formRead it asValue
3 × 33²3 squared9
2 × 2 × 22³2 cubed8
5 × 5 × 5 × 55⁴5 to the fourth625
10 × 10 × 10 × 10 × 1010⁵10 to the fifth100 000
77¹7 to the first7
Any number to the exponent 1 is just that number.

Watch out!

2⁵ is not 2 × 5. 2 × 5 = 10, but 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. The exponent is a count of factors, not a factor itself.

Worked example

Example 1: Find the value of a power

Find the value of 3⁴.

  1. The base is 3 and the exponent is 4, so write 3 four times: 3 × 3 × 3 × 3.
  2. Multiply step by step: 3 × 3 = 9, then 9 × 3 = 27, then 27 × 3 = 81.

3⁴ = 81

Worked example

Example 2: Write repeated multiplication as a power

Write 6 × 6 × 6 in exponential form and find its value.

  1. The factor 6 is used 3 times, so the base is 6 and the exponent is 3: 6³.
  2. 6 × 6 = 36, then 36 × 6 = 216.

6 × 6 × 6 = 6³ = 216 ("6 cubed")

Worked example

Example 3: Prime factorization with exponents

Write the prime factorization of 72 using exponents.

  1. Make a factor tree: 72 = 8 × 9.
  2. 8 = 2 × 2 × 2 and 9 = 3 × 3. So 72 = 2 × 2 × 2 × 3 × 3.
  3. Three 2s make 2³ and two 3s make 3². So 72 = 2³ × 3².
  4. Check: 2³ = 8, 3² = 9, and 8 × 9 = 72. ✓

72 = 2³ × 3²

Think like a mathematician

When you find the value of a power, write the repeated multiplication first, then multiply one step at a time and keep a running total. Big powers grow fast, so check whether your answer makes sense: 2¹⁰ is a little more than 1 000, not 20.

Hands-on activity

Paper-folding powers

You need: a sheet of paper (newspaper or printer paper), a pencil.

  1. Make a table with two columns: Number of folds and Number of layers.
  2. Fold the paper in half. Count the layers (look at the edge). Record it.
  3. Keep folding in half and recording. Predict each answer before you count.
  4. Write each number of layers as a power of 2. For example, after 3 folds: 8 = 2³.
  5. You will probably find it very hard to fold more than 6 or 7 times. Use exponents to predict how many layers there would be after 10 folds.

What to look for: layers 2, 4, 8, 16, 32, 64, 128... After n folds there are 2ⁿ layers. After 10 folds: 2¹⁰ = 1 024 layers.

Math talk: "Why does the exponent match the number of folds?"

Just for fun

Fold a sheet of paper in half just 7 times and it already has 2⁷ = 128 layers. That is why the 8th fold feels almost impossible. Paper: 1, You: 0.

Your turn

1

Write 4 × 4 × 4 in exponential form.

2

Find the value of 2⁶.

3

Find the value of 5².

4

Find the value of 10⁴.

5

Which is greater: 2⁵ or 5²?

6

Write 7⁴ as repeated multiplication and find its value.

7

In 9³, name the base and the exponent. Then find the value.

More practice: warm-up

1

Find the value of 3².

2

Find the value of 1⁸.

3

Find the value of 6².

4

Find the value of 10³.

5

Write 8 × 8 in exponential form.

More practice: on your own

1

Find the value of 3⁵.

2

Find the value of 2³ × 3².

3

Write the prime factorization of 200 using exponents.

4

A cube-shaped storage box has edges of 4 dm. What is its volume? Use an exponent.

5

Which is greater: 4³ or 3⁴? Show how you know.

Challenge

1

On a snow day, a school uses a phone tree. The principal calls 3 families. Each of those families calls 3 new families, and so on. How many families get a call in the 4th round? How many families are called altogether in the 4 rounds?

2

If 2ⁿ = 1 024, what is the exponent n?

3

Which is greater, 2¹⁰ or 10³? By how much?

Check your understanding

Quiz

1.

What is the value of 2⁴?

2.

Which is the same as 5 × 5 × 5?

3.

In the power 7², what is the base?

4.

What is 10⁵?

5.

Which is equal to 3⁴?

6.

What is the value of 2³ × 5?

For parents and teachers

Big idea: The Ontario curriculum (B2.7) asks Grade 7 students to evaluate and express repeated multiplication of whole numbers using exponential notation, in various contexts. This is the first year exponents appear, so go slowly with the vocabulary: base, exponent, power.

Common difficulties:

  • Multiplying the base by the exponent (2⁵ = 10). Have your child always write out the repeated multiplication first.
  • Mixing up the base and exponent (thinking 3⁴ = 4³). Compare the values to see they differ.
  • Losing track in long products. Encourage a running total: 3, 9, 27, 81.

How to help: Do the paper-folding activity together, and spot squares and cubes at home (floor tiles in a square, ice-cube trays, dice). This lesson also builds A1 skills: noticing patterns, making predictions and checking them.