How to Multiply Exponents? (+FREE Worksheet!)
Multiplication Property of Exponents
When you multiply powers that share the same base, you simply add the exponents: \(a^m \cdot a^n = a^{m+n}\). One short rule replaces a lot of repeated multiplying. Let’s see why it works and drill it, with a solver, practice, and a worksheet maker a tap away.
Laws of Exponents: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Check the baseMake sure the repeated factor is the same.
- Match the operationMultiplication, division, and powers of powers use different exponent moves.
- Clean negativesMove negative exponents across the fraction bar and make them positive.
Worked examples
Multiply same bases
- The base is x in both powers.
- Multiplication means add exponents.
- 3 + 4 = 7.
Power of a power
- The whole power is raised to another power.
- Multiply the exponents.
- 2 times 5 is 10.
Try one before moving on
Laws of Exponents: pop-up practice
The multiplication property of exponents says that to multiply two powers with the same base, you keep the base and add the exponents: \(a^m \cdot a^n = a^{m+n}\). That one shortcut turns long repeated multiplication into a single step, and it’s what makes polynomials and scientific notation manageable.
For example, \(x^3 \cdot x^4 = x^{7}\), three \(x\)’s multiplied by four more make seven in all.
Why You Add the Exponents
An exponent counts how many times the base is multiplied by itself. So \(x^3 \cdot x^4\) is \((x\cdot x\cdot x)(x\cdot x\cdot x\cdot x)\), seven \(x\)’s in a row, which is \(x^7\). You didn’t really “add” by magic; you just counted all the factors. That’s why the rule is add, not multiply, the exponents.
How to multiply powers (same base):
- Confirm the bases are identical.
- Keep that base.
- Add the exponents.
Handling Coefficients and Several Variables
Add exponents
\(a^2 \cdot a^5 = a^{7}\)
Multiply numbers, add exponents
Numbers multiply; matching variables add.
Group by base
Add exponents within each base separately.
Worked Examples
Write out the factors and you can see why the exponents add, traced on each card.
Example A, Same base
Simplify \(x^3 \cdot x^4\).
- \(x^3\) is three \(x\)’s, \(x^4\) is four more.
- Together that’s seven \(x\)’s multiplied.
- Add the exponents: \(3 + 4 = 7\), so \(x^7\).
Answer: \(x^{7}\)
Example B, A numeric base
Simplify \(2^2 \cdot 2^3\).
- The base is the same, so add exponents: \(2+3 = 5\).
- The base stays 2, not 4.
- \(2^5 = 32\).
Answer: 32
Example C, With coefficients
Simplify \(3x^2 \cdot 4x^3\).
- Multiply the coefficients: \(3 \cdot 4 = 12\).
- Add the exponents: \(2 + 3 = 5\).
- Combine: \(12x^5\).
Answer: \(12x^{5}\)
Example D, Two variables
Simplify \(x^2y^3 \cdot x^4y\).
- Add exponents per base, separately.
- \(x^{2+4} = x^6\) and \(y^{3+1} = y^4\) (remember \(y = y^1\)).
- Combine: \(x^6 y^4\).
Answer: \(x^{6}y^{4}\)
Exponents in the Wild
This rule is what makes scientific notation work. Multiplying \((3\times10^4)(2\times10^5)\) means \(3\cdot2 = 6\) and \(10^{4+5}=10^9\), so \(6\times10^9\). Computer storage uses powers of \(2\), a kilobyte is \(2^{10}\) bytes, and area-times-length volume calculations lean on the same “add the exponents” move.
Slip-Ups That Cost Easy Points
- Multiplying the exponents. \(x^3 \cdot x^4\) is \(x^7\), not \(x^{12}\). Same-base multiplication adds exponents.
- Multiplying the bases. \(2^2 \cdot 2^3 = 2^5\), not \(4^5\). The base is kept, never multiplied by itself.
- Combining different bases. \(x^2 \cdot y^3\) can’t be combined, the rule needs the same base.
- Forgetting an invisible exponent of 1. \(y\) is \(y^1\), so \(y \cdot y^6 = y^7\).
- Confusing it with addition. \(x^3 + x^4\) does not become \(x^7\). You only add exponents when the powers are multiplied, never when they’re added.
Your Turn: Simplify
Use the rule, then reveal the answers. Stuck? The exponent solver shows each step.
- \(x^5 \cdot x^2\)
- \(y \cdot y^6\)
- \(4^2 \cdot 4^2\)
- \(2^3 \cdot 2^4\)
- \(m^3 \cdot m^3\)
- \(5x^2 \cdot 3x^4\)
- \(2x^2y \cdot 5xy^3\)
Show answers
- \(\color{blue}{x^{7}}\)
- \(\color{blue}{y^{7}}\)
- \(\color{blue}{4^{4}=256}\)
- \(\color{blue}{2^{7}=128}\)
- \(\color{blue}{m^{6}}\)
- \(\color{blue}{15x^{6}}\)
- \(\color{blue}{10x^{3}y^{4}}\)
Make Your Own Exponents Worksheet
Generate fresh exponent problems with a full answer key, print or save as a PDF.
Frequently Asked Questions
What is the multiplication property of exponents?
The multiplication property, or product rule, says that when you multiply powers with the same base, you keep the base and add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ. For example, 2³ × 2⁴ = 2⁷ = 128. It works because 2³ is three 2s and 2⁴ is four 2s, so together they make seven 2s multiplied.
What do you do with exponents when multiplying?
If the bases are the same, add the exponents. If the bases are different, the exponents stay as they are. Coefficients are multiplied normally. So 4x² × 3x⁵ = 12x⁷, while x³ × y² stays x³y². A different rule applies to a power raised to a power, like (x³)², where you multiply the exponents to get x⁶ instead of adding them.
How do you solve multiplication with exponents?
Group the numbers and each variable, then work one group at a time. For (−2a³b)(5a²b⁴), multiply the coefficients: −2 × 5 = −10. Then add exponents for each base: a³⁺² = a⁵ and b¹⁺⁴ = b⁵. The answer is −10a⁵b⁵. Remember that a variable with no exponent shown, like b, has an exponent of 1.
What are the multiplication properties of exponents?
Three exponent rules involve multiplication. The product of powers rule adds exponents: x² × x⁶ = x⁸. The power of a power rule multiplies them: (x²)³ = x⁶. The power of a product rule gives the exponent to every factor: (3x)² = 9x². Knowing which situation you are in is the whole trick, since adding when you should multiply is the most common mistake.
What are some examples of the multiplication property of exponents?
Here are a few to try: 5² × 5 = 5³ = 125; y⁴ × y⁴ = y⁸; 10³ × 10⁶ = 10⁹, which is one billion; and 2m³ × 7m⁻¹ = 14m². The last one shows the rule also works with negative exponents, since 3 + (−1) = 2. A quick check for any example is to expand the powers and count the factors.
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