Enter three monomial factors
Each outer power applies to the complete factor. Whole-number exponents from -20 to 20 are accepted. Outer powers range from 0 to 12.
Rules behind the simplification
For the default expression, (3xy)³ becomes 27x³y³. Then multiply 4, 2, and 27. Add matching x exponents. Add matching y exponents.
Get an accurate simplified monomial
- Enter a coefficient, x exponent, y exponent, and outer power for each factor.
- Use an exponent of zero when a variable does not appear.
- Choose Simplify product to place the answer above the form.
- Review the expanded factors and exponent totals in the worked solution.
- Export the current solution as a CSV file or PDF summary.
Default expression breakdown
| Factor | Coefficient | x exponent | y exponent | Outer power | After the power |
|---|---|---|---|---|---|
| (4xy) | 4 | 1 | 1 | 1 | 4xy |
| (2x²y) | 2 | 2 | 1 | 1 | 2x²y |
| (3xy)³ | 3 | 1 | 1 | 3 | 27x³y³ |
| Final product | 216 | 6 | 5 | — | 216x⁶y⁵ |
Why Monomial Products Need Careful Simplification
Multiplying monomials becomes easy when every part follows a clear rule. A monomial has a numerical coefficient and variable factors. The expression (4xy)(2x²y)(3xy)³ contains both features. It also includes an outer power. That power affects every factor inside its parentheses.
Start with the third factor. Raise 3xy to the third power. The coefficient becomes 3³, which equals 27. The x exponent becomes three. The y exponent also becomes three. The factor is therefore 27x³y³. This expansion prevents a common error. Do not multiply only the coefficient by the outer power.
Combine Coefficients and Exponents
Next multiply the numerical coefficients. The values are 4, 2, and 27. Their product is 216. Then combine the x factors. Their exponents are 1, 2, and 3. Addition gives 6. Combine y in the same way. Its exponents are 1, 1, and 3. Addition gives 5.
The simplified answer is 216x⁶y⁵. Standard algebra notation removes multiplication signs between the coefficient and variables. It also omits exponents equal to one. This calculator shows every contribution before producing the final monomial. That structure helps you check classwork, homework, and practice questions.
Use Powers Before Combining Like Bases
Order matters in these problems. Apply any power outside parentheses first. After that, multiply coefficients. Finally, add exponents for matching variable bases. You can use the same pattern with negative coefficients, zero exponents, or more variables. Keep unlike bases separate. For example, x and y cannot combine into xy through exponent addition.
Check the result by comparing factors. The total coefficient should include each powered coefficient. The final x exponent should count every x factor. The final y exponent should do the same. A quick check catches missing powers and incorrect additions.
Build Stronger Algebra Habits
Practice with different values after studying the worked steps. Change one exponent and observe the result. Test a coefficient of one. Try a factor with an outer power of zero. These small experiments show why the rules work. They also make later polynomial multiplication easier. Accurate monomial work supports factoring, equations, graphs, and scientific formulas. Use the export tools to save a record of your examples. Review those records before tests or lessons. Share findings with tutors, classmates, or colleagues when collaborative checking improves confidence.
Common questions about simplifying monomials
1. What is the simplified form of (4xy)(2x²y)(3xy)³?
The result is 216x⁶y⁵. First change (3xy)³ into 27x³y³. Then multiply the coefficients and add exponents of matching variables.
2. Why is 3xy raised to every part of the third factor?
The power of a product distributes across every factor inside parentheses. Therefore, (3xy)³ equals 3³x³y³, or 27x³y³.
3. Why are x exponents added during multiplication?
Matching bases multiply by adding exponents. For example, x¹ times x² times x³ equals x⁶ because 1 + 2 + 3 equals 6.
4. Can I multiply x and y exponents together?
No. x and y are different bases. Add exponents only when bases match. Keep x powers and y powers in separate groups.
5. What happens when a variable exponent is zero?
A nonzero base raised to zero equals one. The variable disappears from that factor, but other matching variable factors can remain in the final answer.
6. Does an outer power affect a negative coefficient?
Yes. An even outer power makes a negative coefficient positive. An odd outer power keeps it negative. The calculator applies this automatically.
7. Can this calculator use negative variable exponents?
Yes. Enter a negative exponent to represent a reciprocal factor. The result may display a negative exponent, which can later be rewritten as a fraction.
8. Why does the coefficient become 216?
The coefficients are 4, 2, and 3³. Since 3³ equals 27, multiplying 4 × 2 × 27 gives 216.
9. What if a factor has an outer power of one?
An outer power of one changes nothing. The factor keeps its original coefficient and variable exponents.
10. How do CSV and PDF exports help?
CSV export creates a spreadsheet-ready record of the expression, coefficient, and exponent totals. PDF export creates a compact solution summary for printing or sharing.
11. Can I use this method for more variables?
Yes. Apply powers to every variable, multiply coefficients, then add exponents for each matching base. Repeat the same process for a, b, z, or other variables.