How to Use Exponents to Write down Multiplication Expressions?

How to use exponents to write down multiplication expressions, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

When you multiply the same number by itself several times, writing it out in full gets long fast: \(\color{blue}{4 \times 4 \times 4 \times 4 \times 4}\). Exponents give you a compact notation for exactly this kind of repeated multiplication. Understanding exponents is an essential building block for algebra, scientific notation, and many GED math problems.

What Is an Exponent?

An exponent (also called a power or index) tells you how many times to use the base as a factor in a multiplication.

bⁿ = \(\color{blue}{b \times b \times b}\) × … (n times)

In the expression \(\color{blue}{4^{3}}\):

• 4 is the base (the number being multiplied)
• 3 is the exponent (how many times 4 is used as a factor)
• \(\color{blue}{4^{3} = 4 \times 4 \times 4 = 64}\)

How to Write Multiplication Expressions Using Exponents

Step 1, Identify the repeated factor

Look at the multiplication expression and find the number being multiplied by itself.

Example: \(\color{blue}{5 \times 5 \times 5 \times 5}\), the repeated factor is 5.

Step 2, Count the factors

Count how many times that factor appears. That count is the exponent.

Example: \(\color{blue}{5 \times 5 \times 5 \times 5}\), the factor 5 appears 4 times. \(\color{blue}{\text{ Exponent } = 4}\).

Step 3, Write in exponential form

Write the base with the exponent in superscript.

Example: \(\color{blue}{5 \times 5 \times 5 \times 5 = 5^{4}}\)

Special exponents to know

• Exponent of 1: Any number to the power of 1 equals itself: \(\color{blue}{7^{1} = 7}\).
• Exponent of 0: Any nonzero number to the power of 0 equals 1: \(\color{blue}{9^{0} = 1}\).
• Squared: exponent of 2 (e.g., \(\color{blue}{6^{2} = 36}\)).
• Cubed: exponent of 3 (e.g., \(\color{blue}{4^{3} = 64}\)).

Step-by-Step Summary

1. Find the repeated base in the multiplication expression.
2. Count how many times it appears. That is the exponent.
3. Write \(\color{blue}{\text{ base }^{\text{ exponent }}}\).
4. To expand back: write the base as a factor (exponent) times.

Watch: What Is an Exponent? (Video Lesson)

Math with Mr. J gives a clear introduction to exponents and what they mean:

Worked Examples

Example 1: Write \(\color{blue}{4 \times 4 \times 4}\) using an exponent and evaluate.

Base: 4, repeated 3 times. Exponential form: \(\color{blue}{4^{3}}\). Value: \(\color{blue}{4 \times 4 \times 4 = 64}\).
Answer: 4³ = 64

Example 2: Write \(\color{blue}{2 \times 2 \times 2 \times 2 \times 2}\) using an exponent and evaluate.

Base: 2, repeated 5 times. Exponential form: \(\color{blue}{2^{5}}\). Value: \(\color{blue}{2 \times 2 \times 2 \times 2 \times 2 = 32}\).
Answer: 2⁵ = 32

Example 3: Expand \(\color{blue}{3^{4}}\) and evaluate.

Expand: \(\color{blue}{3 \times 3 \times 3 \times 3}\). Value: \(\color{blue}{9 \times 9 = 81}\).
Answer: 81

Example 4: Evaluate \(\color{blue}{5^{2}}\).

\(\color{blue}{5 \times 5 = 25}\).
Answer: 25

More Practice: Math Antics, Intro to Exponents

Math Antics explains the concept of exponents and indices with visual examples:

Exercises

1. Write \(\color{blue}{6 \times 6 \times 6}\) using an exponent. Evaluate.
2. Expand \(\color{blue}{2^{6}}\) and evaluate.
3. Write \(\color{blue}{10 \times 10 \times 10 \times 10}\) using an exponent. Evaluate.
4. Evaluate \(\color{blue}{7^{2}}\).
5. Write the expression “three to the fifth power” in exponential form and evaluate.
6. Evaluate \(\color{blue}{1^{50}}\).

Answers

1. \(\color{blue}{6^{3} = 216}\)
2. \(\color{blue}{2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64}\)
3. \(\color{blue}{10^{4} = 10,000}\)
4. \(\color{blue}{49}\)
5. \(\color{blue}{3^{5} = 243}\)
6. \(\color{blue}{1}\)

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Frequently Asked Questions

Is an exponent repeated multiplication?

Yes. A positive whole-number exponent tells you how many times to use the base as a factor. So 4³ means 4 × 4 × 4, which equals 64. The base is the number being multiplied, and the exponent is the count of how many times it appears in the product.

How do you write repeated multiplication in exponential form?

Find the factor that repeats, count how many times it appears, and write that count as a small raised number next to it. For 5 × 5 × 5 × 5, the factor 5 appears 4 times, so it becomes 5⁴. With a variable it works the same way: y × y × y = y³.

What is repeated multiplication called?

Repeated multiplication of the same number is called exponentiation, or raising a number to a power. The result is called a power. For example, 2 × 2 × 2 × 2 × 2 is written 2⁵ and read “2 to the fifth power,” and its value is 32. A power of 2 is called squared and a power of 3 is called cubed.

Are exponents repeated addition?

No. Exponents are repeated multiplication, while multiplication is repeated addition. Mixing them up is a common mistake: 3⁴ means 3 × 3 × 3 × 3 = 81, not 3 × 4 = 12 or 3 + 3 + 3 + 3 = 12. Writing out the factors before you calculate helps avoid this error.

How do you multiply exponents?

When two powers have the same base, keep the base and add the exponents. This works because you are just counting all the repeated factors. For example, 2³ × 2⁴ = 2⁷, since three 2s times four 2s is seven 2s, and 2⁷ = 128. If the bases are different, such as 2³ × 3², find each value first: 8 × 9 = 72.

Related Topics

• Using Exponents to Write Powers of Ten
• Prime Factorization with Exponents
• Evaluate Integers Raised to Rational Exponents
• Order of Operations
• Identifying Expressions and Equations