Can You Multiply Numbers with Different Exponents? | Rule That Sticks

Exponents trip people up because two different ideas get mixed together. One is multiplying powers that share the same base, like 2³ × 2⁵. The other is multiplying powers that have different bases, like 2³ × 3⁵. Those are not the same move, and that’s where most mistakes start.

Here’s the clean way to think about it: an exponent belongs to its own base. It doesn’t float across the whole expression unless the power is outside parentheses. So when you multiply numbers with different exponents, you do not mash the exponents together unless a rule says you can. Most of the time, you either multiply the values directly or rewrite the expression into matching bases first.

Can You Multiply Numbers With Different Exponents?

Yes, but the rule depends on the bases. If the bases match, add the exponents. If the bases do not match, leave the exponents attached to their own bases and multiply the terms as written. That one split keeps the whole topic tidy.

Take these two examples:

• 2³ × 2⁴ = 2⁷
• 2³ × 3⁴ = 8 × 81 = 648

In the first line, the base is 2 in both terms, so the exponents add. In the second line, the bases are different, so there is no “add the exponents” move. You either evaluate each power, or keep the product in factored form if that’s cleaner for the job at hand.

When Exponents Add And When They Do Not

The product rule is short and strict: a^m × aⁿ = a^m+n. That rule works only when the base is the same. OpenStax states the product rule in that exact form in its algebra text, and it’s the piece most students need to lock in early. You can see it in OpenStax’s section on exponents and scientific notation.

Once the bases change, the rule changes too. Multiplying 5² × 7³ is not 35⁵. That answer looks neat, but it breaks the structure of exponents. The 2 belongs to the 5. The 3 belongs to the 7. They stay attached unless a larger rule, such as a power outside parentheses, tells you to distribute that exponent across a product.

Same Base

These are the easy wins:

• x⁴ × x² = x⁶
• 10³ × 10⁶ = 10⁹
• y × y⁵ = y⁶

That last line shows a sneaky point: a plain y is really y¹. Once you spot that, the addition step feels natural.

Different Base

These need a different move:

• 2² × 3² = (2 × 3)² = 6² = 36
• 2³ × 3² = 8 × 9 = 72
• a²b⁵ stays as a²b⁵ unless you have values for a and b

Notice the first line. The exponents match, so you can use the rule (ab)ⁿ = aⁿbⁿ in reverse. That does not mean the bases suddenly match. It means the shared exponent lets you group the factors neatly.

How To Work Through These Problems Without Guessing

A steady method beats memory tricks. Run through the expression in this order:

1. Check whether the bases match.
2. If they match, add exponents.
3. If they do not match, check whether the exponents match.
4. If the exponents match, you may combine the bases inside one set of parentheses.
5. If neither matches, keep each power attached to its own base and simplify from there.

Khan Academy’s exponent properties review lays out these patterns in student-friendly form, especially the split between product of powers and power of a product. It’s a handy reference when the symbols start to blur together: exponent properties review.

That checklist also helps with algebra terms. In 3x² × 4x⁵, multiply the coefficients first: 3 × 4 = 12. Then use the same-base rule on the x terms: x² × x⁵ = x⁷. Final answer: 12x⁷.

In 3x² × 4y⁵, the variables are different. So you multiply the coefficients and keep the powers attached: 12x²y⁵.

Expression	What You Check	Result
23 × 24	Same base	27
52 × 72	Same exponent	(5 × 7)2 = 352
23 × 34	Different base, different exponent	8 × 81 = 648
x6 × x	x = x1	x7
3x2 × 4x5	Multiply coefficients, then same base	12x7
3x2 × 4y5	Different variables	12x2y5
(ab)4	Power on a product	a4b4
a-2 × a5	Add exponents, including negatives	a3

Cases That Cause The Most Mistakes

When The Exponents Match But The Bases Do Not

This is the spot where students often freeze. If you see 4³ × 5³, the shared exponent lets you write (4 × 5)³ = 20³. That works because both factors are raised to the same power.

But 4³ × 5² does not turn into 20⁵. There is no rule for that. You evaluate or leave it factored.

When A Power Sits Outside Parentheses

Parentheses change the game. In (2x)⁴, the exponent applies to both 2 and x. So you get 2⁴x⁴ = 16x⁴. OpenStax walks through this under the power rule and related exponent properties in its algebra material on integer exponents and scientific notation.

Compare that with 2x⁴. Here, only x is raised to the fourth power. The 2 stays plain. One pair of parentheses can flip the whole answer, so read the expression before you start simplifying.

When Negative Exponents Show Up

Negative exponents do not block multiplication. They just change where the factor belongs in the final form. Since a^-n = 1 / aⁿ, you can still use the same-base rule first.

• a^-3 × a⁵ = a²
• x⁴ × x^-6 = x^-2 = 1 / x²

That last rewrite matters in classwork and many textbooks, since answers are often expected with positive exponents only.

Algebra Examples That Feel Closer To Real Homework

Let’s run through a few mixed expressions the way you’d see them on a worksheet.

Example 1

6m³ × 2m⁴

Multiply coefficients: 6 × 2 = 12. Then add exponents on m: m³ × m⁴ = m⁷. Final answer: 12m⁷.

Example 2

5a²b³ × 3ab⁴

Multiply coefficients: 15. Then combine like bases: a² × a = a³, and b³ × b⁴ = b⁷. Final answer: 15a³b⁷.

Example 3

(3x²y)²

Square each factor: 3²x⁴y² = 9x⁴y². Here the outside exponent hits every factor inside the parentheses.

Common Mistake	Wrong Move	Correct Move
23 × 34	67	8 × 81 = 648
(2x)3	2x3	8x3
x2 × x-5	x7	x-3 = 1/x3
42 × 52	204	202

One Memory Trick That Actually Helps

Ask one question: “Who owns the exponent?” If the exponent belongs to the same base on both sides, you can add. If the exponent sits outside parentheses, it reaches every factor inside. If neither of those is true, leave the powers attached to their own bases.

That small check stops most slipups before they happen. It also keeps you from chasing fake shortcuts that look slick on paper and fall apart on the next line.

What To Write On A Test

If the problem says simplify, your safest route is this:

• Combine coefficients.
• Combine like bases by adding exponents.
• Use a shared outside exponent only when parentheses show it.
• Rewrite negative exponents with positive exponents if needed.

That’s the whole pattern. Once you separate “same base” from “same exponent,” multiplying numbers with different exponents stops feeling random. The rules are strict, but they’re also tidy, and that makes this one of those algebra topics that gets easier fast after a few clean reps.