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
Why do you add exponents when multiplying?
Because an exponent counts repeated factors. \(x^3 \cdot x^4\) lines up 3 then 4 copies of \(x\) — 7 in all — so the result is \(x^7\). Adding the exponents just counts the total factors.
Does the base change?
No. \(2^2 \cdot 2^3 = 2^5\); the base stays 2. You only add the exponents — never multiply the bases.
What if the bases are different?
The rule doesn’t apply. \(x^2 \cdot y^3\) stays as it is, because the bases (\(x\) and \(y\)) aren’t the same.
How do coefficients work?
Multiply the coefficients normally and add the exponents of matching variables: \(3x^2 \cdot 4x^3 = 12x^5\).
Related Topics
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