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
Continue Your Study
Ready for the next step? Pick up right where this lesson leaves off:
Related to This Article
More math articles
- 7th Grade Georgia Milestones Assessment System Math Worksheets: FREE & Printable
- How to Do Division Using Partial Quotients
- The Ultimate GED Math Formula Cheat Sheet for 2026!
- 4th Grade NDSA Math Worksheets: FREE & Printable
- The Best Algebra 1 Book for Massachusetts Students
- Grade 8 ELA Tests: A Clear Parent Guide to Reading, Writing, and Test-Day Readiness
- Polygon Names
- District of Columbia Algebra 1 Free Worksheets: 64 Free Printable Algebra 1 Practice PDFs
- Building a Well-Organized Paragraph
- Logrolling
What people say about "How to Multiply Exponents"?
No one replied yet.