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.

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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

What is the difference between the base and the exponent?

The base is the number being multiplied (the factor). The exponent tells you how many times to multiply that base by itself. In (color{blue}{5^{3}}), 5 is the base and 3 is the exponent.

Does the order of base and exponent matter?

Yes, it matters a lot. (color{blue}{2^{3} = 8}) but (color{blue}{3^{2} = 9}). The base and exponent cannot be swapped (except in special cases where they give the same result, like (color{blue}{2^{2} = 4 = 4^{1}})).

What does a negative exponent mean?

A negative exponent means take the reciprocal of the base raised to the positive exponent: (color{blue}{2^{-3} = frac{1}{2^{3}} = frac{1}{8}}). This is covered in more advanced exponent lessons.

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