Product of Powers
Multiplying powers with the same base adds the exponents.
Formula
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Example
2³ · 2⁴ = 2⁷ = 128.
How it works
When two powers have the same base and are multiplied, you keep the base and combine the exponents by adding them. This works because each exponent tells how many times the base is used as a factor, so multiplying just puts all of those repeated factors together. For example, a^m means a is multiplied by itself m times, and a^n means a is multiplied by itself n times, so together that makes m+n copies of a.
When to use it
Use this rule any time you are multiplying expressions like x^3 · x^5 or 10^2 · 10^4. It is common in simplifying algebraic expressions, scientific notation, and exponent problems before doing more advanced calculations.
Worked example, step by step
Start with: 5^2 · 5^3 Use the product of powers rule: 5^(2+3) Add the exponents: 5^5 Evaluate: 5 × 5 × 5 × 5 × 5 = 3125 Final answer: 5^2 · 5^3 = 5^5 = 3125
Common mistakes to avoid
- Adding the bases instead of the exponents, such as writing 5^2 · 5^3 = 10^5
- Using this rule when the bases are different, like trying to combine 2^3 · 3^3 the same way
- Multiplying the exponents instead of adding them, such as writing a^m · a^n = a^(mn)
Did You Know?
Adding exponents when multiplying powers is the rule that makes logarithms possible.
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