How to Multiply Exponents? (+FREE Worksheet!)
The multiplication property of exponents is one of the most-used rules in algebra. Once you see the pattern, multiplying powers with the same base becomes automatic. This guide explains the rule, shows you how it works step by step, and gives you plenty of practice with fully solved examples and two short video lessons.
What Is the Multiplication Property of Exponents?
When you multiply two powers that share the same base, you keep the base and add the exponents. In symbols:
am × an = a\(\color{blue}{m + n}\)
The base \(\color{blue}{a}\) can be any number or variable. The exponents \(\color{blue}{m}\) and \(\color{blue}{n}\) must be integers (positive, negative, or zero).
How to Apply the Rule
1. Same base — add exponents
Identify that both factors share the same base, then simply add the exponents and write the result as one power.
- \(\color{blue}{x^{3} \times x^{5} = x^{3 + 5} = x^{8}}\)
- \(\color{blue}{2^{3} \times 2^{4} = 2^{3 + 4} = 2^{7} = 128}\)
2. Coefficients multiply normally
When there are numerical coefficients in front of the variable bases, multiply the coefficients and add the exponents separately.
- \(\color{blue}{(3x^{2})(4x^{5}) = (3 \times 4) \times x^{2 + 5} = 12x^{7}}\)
- \(\color{blue}{(2y^{4})(5y^{3}) = 10y^{7}}\)
3. Different bases — do NOT add exponents
The rule only works when the bases are identical. \(\color{blue}{x^{3} \times y^{5}}\) cannot be simplified further because \(\color{blue}{x}\) and \(\color{blue}{y}\) are different bases.
Step-by-Step Summary
- Check that both powers have the same base.
- Keep the base exactly as it is.
- Add the two exponents to form the new exponent.
- If there are coefficients, multiply them separately.
- Write the simplified expression.
Watch: Multiplication Property of Exponents (Video Lesson)
This Khan Academy lesson covers the product-of-powers property with clear numeric and variable examples:
Multiplication Property of Exponents – Worked Examples
Example 1: Simplify \(\color{blue}{y^{4} \times y^{2}}\).
Same base \(\color{blue}{y}\). Add exponents: \(\color{blue}{4 + 2 = 6}\).
Answer: \(\color{blue}{y^{4} \times y^{2} = y^{6}}\)
Example 2: Simplify \(\color{blue}{3^{2} \times 3^{3}}\).
Same base \(\color{blue}{3}\). Add exponents: \(\color{blue}{2 + 3 = 5}\). Then evaluate: \(\color{blue}{3^{5} = 243}\).
Answer: \(\color{blue}{3^{2} \times 3^{3} = 3^{5} = 243}\)
Example 3: Simplify \(\color{blue}{(5x^{3})(2x^{6})}\).
Multiply coefficients: \(\color{blue}{5 \times 2 = 10}\). Add exponents: \(\color{blue}{3 + 6 = 9}\).
Answer: \(\color{blue}{(5x^{3})(2x^{6}) = 10x^{9}}\)
Example 4: Simplify \(\color{blue}{a^{2} \times a^{5} \times a^{3}}\).
Three factors, same base \(\color{blue}{a}\). Add all exponents: \(\color{blue}{2 + 5 + 3 = 10}\).
Answer: \(\color{blue}{a^{2} \times a^{5} \times a^{3} = a^{10}}\)
More Practice: Step-by-Step Video Review
For additional worked examples covering both power and multiplication properties together:
Exercises for the Multiplication Property of Exponents
Simplify each expression.
- \(\color{blue}{a^{4} \times a^{6}}\)
- \(\color{blue}{5^{2} \times 5^{3}}\)
- \(\color{blue}{b^{7} \times b^{0}}\)
- \(\color{blue}{2^{4} \times 2^{5}}\)
- \(\color{blue}{x^{3} \times x^{4} \times x^{2}}\)
- \(\color{blue}{4^{1} \times 4^{3}}\)
Answers
- \(\color{blue}{a^{10}}\)
- \(\color{blue}{5^{5} = 3125}\)
- \(\color{blue}{b^{7}}\)
- \(\color{blue}{2^{9} = 512}\)
- \(\color{blue}{x^{9}}\)
- \(\color{blue}{4^{4} = 256}\)
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Frequently Asked Questions
What does the multiplication property of exponents say?
When multiplying two powers with the same base, keep the base and add the exponents: \(\color{blue}{a^{m} \times a^{n} = a^{m + n}}\).
Does the rule work with negative exponents?
Yes. For example, \(\color{blue}{x^{3} \times x^{-1} = x^{3 + (-1)} = x^{2}}\). Just add the signed exponent values.
What if the bases are different?
The rule does not apply. \(\color{blue}{x^{3} \times y^{4}}\) stays as \(\color{blue}{x^{3}y^{4}}\) because the bases \(\color{blue}{x}\) and \(\color{blue}{y}\) are different.
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