When can I add exponents, and when should I multiply them?
Quick answer
Add exponents when you multiply powers with the same base: 5³ · 5⁴ = 5³⁺⁴ = 5⁷. Multiply exponents when one power is raised to another power: (5³)⁴ = 5³×⁴ = 5¹². [S1] First look for the operation between the powers: multiplication or a power raised to a power. The rules are not interchangeable.
The question, the details, the next step.
I keep mixing up whether exponents add or multiply. What should I look at first?
Look at the expression’s structure—not just at the exponents.
- If you are multiplying powers with the same base, add their exponents: aᵐ · aⁿ = aᵐ⁺ⁿ. [S1]
- If a power is raised to another power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ. [S1]
In the first pattern, two powers of the same base are multiplied together. In the second, the entire inner power is being raised to an outer power. The patterns look similar, but the operations are different.
Can you compare them with the same numbers so I can see the difference?
Here is a side-by-side comparison using base 5:
Multiply powers with the same base — add exponents 5³ · 5⁴ = 5³⁺⁴ = 5⁷
Check: 5³ is 125 and 5⁴ is 625. Their product is 78,125, which is 5⁷.
Raise a power to a power — multiply exponents (5³)⁴ = 5³×⁴ = 5¹²
Check: 5³ is 125, so the expression is 125⁴, or 5¹².
The textbook gives these same examples to contrast the rules. [S1] The quick comparison is: a product between same-base powers means add; a power raised to another power means multiply.
What if I use the wrong rule? How can I catch that before moving on?
Try a small-number check. For example, compare 2³ · 2² with (2³)².
For the product, add: 2³ · 2² = 2⁵ = 32. Directly, that is 8 · 4 = 32.
For the power raised to a power, multiply: (2³)² = 2⁶ = 64. Directly, that is 8² = 64.
If you accidentally use the product rule on the second expression, you might get 2⁵, or 32, which does not match the direct calculation. A check like this helps reveal a rule mix-up; it does not replace reading the expression carefully.
Can you give me a short routine I can use on homework?
Use this routine:
- Mark the operation. Is there multiplication between the powers, or is a power raised to an exponent?
- Check the bases. The add-exponents pattern here is for powers with the same base. [S1]
- Choose the matching rule. Same-base product: add. Power raised to a power: multiply. [S1]
- Keep the base and simplify the exponent arithmetic.
- Check with small numbers if the result seems surprising.
Copyable note: “I see ____. The base(s) are ____. This matches the ____ rule, so I ____ the exponents. Result: ____.”
For instance: “I see 3² · 3⁵. The bases match and the powers are multiplied, so I add: 3⁷.” These two patterns do not establish rules for every expression involving exponents, such as expressions with different bases.
A practical checklist
Identify whether the powers are multiplied together or one power is raised to another.
For a product, confirm the powers have the same base before adding exponents.
For a power raised to a power, multiply the exponents.
Simplify the exponent arithmetic while keeping the base.
Check a doubtful result by evaluating a small numerical example.
A few more useful answers
Do I add exponents in 2³ · 3⁴?
No—not using the same-base product rule shown here, because the bases differ. This guide covers multiplying powers with the same base and raising a power to a power.
Do parentheses always mean I multiply exponents?
No. The multiply-exponents rule applies to a power raised to another power, as in (aᵐ)ⁿ. Check what operation is being applied to the whole inner power.
How can I remember which rule is which?
Say the operation aloud: same-base powers multiplied together, add; a power raised to a power, multiply. Then use a small-number check if you are unsure.
Sources
References used for this guide.
- [S1] OpenStax College Algebra 2e, Section 1.2: Exponents and Scientific Notation openstax.org · Checked