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B2.7 · Express repeated whole-number multiplication using exponents

Learn to express repeated whole-number multiplication using exponents through clear examples and targeted practice.

Ontario Grade 7 Mathematics

Number

Ontario Grade 7 Mathematics — B2.7

You already know how to multiply whole numbers. For example, 4×44 \times 4 means that two factors, both equal to 4, are multiplied. When the same number appears as a factor many times, writing every factor can become long. An exponent is a shorter way to record that repeated multiplication. In this lesson, we will connect a concrete array to repeated factors, then write those factors using an exponent. We will focus only on repeated multiplication of the same whole number.

What you will learn

  • Recognize when the same whole number is multiplied by itself repeatedly.
  • Write repeated whole-number multiplication using an exponent.
  • Identify the base and exponent in a power.
  • Explain what the exponent tells you without confusing it with the value of the product.

From equal groups to repeated factors

A factor is a number that is multiplied by another number. In 3×3×33 \times 3 \times 3, each 3 is a factor. The same factor appears three times.
An array can help you picture some repeated multiplication. An array has equal rows and equal columns. A square array has the same number of rows as columns. A square array with 3 rows and 3 columns contains 9 items. It represents 3×33 \times 3.
The array shows one useful case, but repeated multiplication does not have to make a square array. For example, five equal factors can be written as 2×2×2×2×22 \times 2 \times 2 \times 2 \times 2. The important feature is that the factor stays the same.
3×3=93 \times 3 = 9
  • A factor is a number being multiplied.
  • Repeated multiplication uses the same factor more than once.
  • A square array can model a factor multiplied by itself twice.

Use an exponent to record repeated factors

An exponent is a small raised number that tells how many times to use the same factor. For example, in 434^3, the 4 is the base and the 3 is the exponent. The base is the repeated factor. The exponent counts how many times the base is used as a factor.
Read 434^3 as “4 to the exponent 3.” It means 4×4×44 \times 4 \times 4. The exponent does not tell you to multiply 4 by 3. It tells you that there are three factors of 4.
A power is an expression written with a base and an exponent, such as 434^3. The repeated multiplication and its power are two ways to show the same calculation. If asked to express repeated multiplication using exponents, count the factors, put their shared value at the base, and put the number of factors as the exponent.
a×a×⋯×a⏟n factors=an\underbrace{a \times a \times \cdots \times a}_{n\text{ factors}} = a^n
  • The base names the repeated factor.
  • The exponent counts how many times that factor appears.
  • Write the exponent as a small raised number.

A careful way to write a power

First, check that every factor is the same. If the factors are not all the same, they cannot be shortened to one base with one exponent using this idea.
Next, identify the repeated number. This becomes the base. Then count how many times it appears. That count becomes the exponent.
Finally, read the expression back as repeated multiplication. This check helps catch a common error: the exponent counts factors; it is not another factor in the multiplication.
For example, the expression 6×6×6×66 \times 6 \times 6 \times 6 has four factors of 6. Its base is 6 and its exponent is 4, so it is written as 646^4. This lesson is about expressing the repeated multiplication. You can leave the expression in power form when the task asks for that form.
b×b×⋯×b=bkb \times b \times \cdots \times b = b^k
  • Check that the factors match.
  • Use the repeated value as the base.
  • Use the number of factors as the exponent.
  • Read the power back as repeated multiplication to check it.

Practice the meaning, not just the notation

When you see a power, name its two parts and expand it into repeated multiplication. When you see repeated multiplication, count the matching factors and write the power. Moving in both directions shows that you understand what the notation means.
Keep the exponent attached to its base. In 535^3, the base is 5 and the exponent is 3. The expression means three factors of 5, not five factors of 3.
Before you finish, ask: What number is repeated? How many copies of it are multiplied? Does my exponent match that count?
  • A power can be expanded into repeated multiplication.
  • The base and exponent have different jobs.
  • A factor count is not the same as the product's value.

Worked example

Read a square array

A square array has 3 rows with 3 counters in each row. Express the total as repeated multiplication and then as a power.
Three by three array
Three by three arrayA square array has 3 rows and 3 columns, for 9 counters in all. It models two factors of 3.3 × 3 = 93 units on each side

A square array has 3 rows and 3 columns, for 9 counters in all. It models two factors of 3.

  1. Connect the array to multiplication
    There are 3 equal rows, each with 3 counters. The total can be recorded as 3 groups of 3, or as 3 multiplied by 3.
    3×33 \times 3
  2. Count the repeated factors
    The factor 3 appears twice. Use 3 as the base and 2 as the exponent.
    323^2
  3. Check the meaning
    Read the power as two factors of 3. The array contains 9 counters, which agrees with the multiplication.
    3×3=93 \times 3 = 9
Answer: The repeated multiplication is 3×33 \times 3. As a power, it is 323^2.
Check: The exponent is 2 because there are two factors of 3.

Worked example

Write five equal factors

Express 2×2×2×2×22 \times 2 \times 2 \times 2 \times 2 using an exponent.
  1. Find the repeated factor
    Every factor is 2, so 2 is the base.
    22
  2. Count the factors
    There are five copies of 2. The exponent is 5 because it records that count.
    252^5
  3. Read it back
    The power means five factors of 2, so it matches the original expression.
    25=2×2×2×2×22^5 = 2 \times 2 \times 2 \times 2 \times 2
Answer: 252^5
Check: The base is 2, and the exponent is 5 because there are five factors.

Worked example

Expand a power and identify its parts

Write 737^3 as repeated multiplication. State the base and explain what the exponent counts.
  1. Identify the base
    The base is the repeated factor. In this power, the repeated factor is 7.
    77
  2. Use the exponent as a count
    The exponent is 3, so use three factors of 7. It does not mean to multiply 7 by 3.
    7×7×77 \times 7 \times 7
  3. Compare the two forms
    The expanded expression has exactly three factors of 7, so it matches the power.
    73=7×7×77^3 = 7 \times 7 \times 7
Answer: The repeated multiplication is 7×7×77 \times 7 \times 7. The base is 7, and the exponent 3 counts the three factors.
Check: There are three factors of 7 in the expanded expression.

Common mistakes and how to avoid them

Reading 434^3 as 4×34 \times 3.
Correction: The exponent counts how many factors of the base to use. Read 434^3 as 4×4×44 \times 4 \times 4.
Using the number of multiplication signs as the exponent.
Correction: Count the factors, not the multiplication signs. For example, 5×5×55 \times 5 \times 5 has three factors of 5, so it is 535^3.
Writing one power for factors that are different.
Correction: A single base and exponent describe repeated copies of the same factor. Check that every factor matches before writing a power.

Lesson summary

  • An exponent is a raised number that counts repeated factors.
  • The base is the factor that repeats.
  • To write repeated multiplication as a power, use the repeated factor as the base and the number of factors as the exponent.
  • To check a power, expand it and confirm that the base appears the stated number of times.

Check your understanding

Question 1

Which power expresses 8×8×8×88 \times 8 \times 8 \times 8?
  1. 838^3
  2. 484^8
  3. 848^4
  4. 8×48 \times 4
Show answer and explanation
848^4
There are four factors, and each factor is 8. The base is 8 and the exponent is 4.

Question 2

What does 353^5 mean?
  1. Five factors of 3 multiplied together
  2. Three factors of 5 multiplied together
  3. 3 multiplied by 5
  4. Five factors of 3 added together
Show answer and explanation
Five factors of 3 multiplied together
The base 3 is the repeated factor. The exponent 5 counts five copies of that factor.

Question 3

In 6×6×66 \times 6 \times 6, what is the exponent when the expression is written as a power?
  1. 6
  2. 3
  3. 2
  4. 18
Show answer and explanation
3
The factor 6 appears three times, so the exponent is 3. The power is 636^3.

Key terms

Factor
A number that is multiplied by another number.
Base
The repeated factor in a power.
Exponent
The small raised number that tells how many times the base is used as a factor.
Power
An expression written with a base and an exponent.

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