Calculate products, quotients, and powers of monomials by entering coefficients, variables, and exponents to get simplified standard form.
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Multiplying Monomials Formula
A monomial has the form c·x1a₁·x2a₂···xnaₙ, where c is the coefficient and each exponent is a whole number. The calculator uses three exponent rules.
Multiply:
(a*x^m*y^n) * (b*x^p*y^q) = (a*b) * x^(m+p) * y^(n+q)
Divide:
(a*x^m*y^n) / (b*x^p*y^q) = (a/b) * x^(m-p) * y^(n-q)
Power:
(a*x^m*y^n)^k = a^k * x^(m*k) * y^(n*k)
- a, b: numerical coefficients
- x, y: variables (any single letters)
- m, n, p, q: exponents on the variables
- k: integer power applied to the whole monomial
Assumptions: each input is a single monomial, not a sum. Coefficients can be integers, decimals, or simple fractions like 3/4. The denominator coefficient in division cannot be zero. Variables that appear in only one monomial keep their original exponent during multiplication and division. A negative result exponent is reported but flagged, since a true polynomial monomial requires non-negative exponents.
The three calculator modes apply directly to the formulas above. Multiply combines two or more monomials by multiplying coefficients and adding matching exponents. Divide subtracts the denominator's exponents from the numerator's and divides the coefficients. Power raises the coefficient to the chosen integer and multiplies every variable exponent by that integer.
Reference Tables
Use these tables to check your work or recall the rule that applies to a step.
| Rule | Form | Example |
|---|---|---|
| Product | x^m · x^n = x^(m+n) | x^2 · x^3 = x^5 |
| Quotient | x^m / x^n = x^(m-n) | x^7 / x^4 = x^3 |
| Power of a power | (x^m)^k = x^(m·k) | (x^3)^4 = x^12 |
| Power of a product | (x·y)^k = x^k · y^k | (2x)^3 = 8x^3 |
| Zero exponent | x^0 = 1, x ≠ 0 | 5x^0 = 5 |
| Negative exponent | x^(-n) = 1 / x^n | x^(-2) = 1/x^2 |
| Input | How the calculator reads it |
|---|---|
| 3x^2y | coefficient 3, x to the 2, y to the 1 |
| -2xy^3 | coefficient -2, x to the 1, y to the 3 |
| (1/2)a^2b | coefficient 0.5, a to the 2, b to the 1 |
| 4x²y³ | superscripts convert to 4x^2y^3 |
| 3x^2y * -2xy^3, 4z | three monomials separated by * and a comma |
Worked Examples
Example 1. Multiply three monomials. Compute (3x^2y)(-2xy^3)(4z).
Multiply the coefficients: 3 · (-2) · 4 = -24. Add exponents on x: 2 + 1 = 3. Add on y: 1 + 3 = 4. The variable z appears only once with exponent 1. Result: -24x^3y^4z.
Example 2. Divide monomials. Compute 12x^5y^2 ÷ 3x^2y.
Divide coefficients: 12 / 3 = 4. Subtract exponents on x: 5 - 2 = 3. Subtract on y: 2 - 1 = 1. Result: 4x^3y.
Example 3. Raise a monomial to a power. Compute (-2x^3y)^4.
Raise the coefficient: (-2)^4 = 16. Multiply exponents by 4: x^(3·4) = x^12, y^(1·4) = y^4. Result: 16x^12y^4.
Example 4. Quotient with a negative exponent. Compute 6x^2y ÷ 2x^5y.
Coefficient: 6 / 2 = 3. Exponent on x: 2 - 5 = -3. Exponent on y: 1 - 1 = 0, so y drops out. Result: 3x^(-3), or 3/x^3.
FAQ
Can I enter more than two monomials at once? Yes. In Multiply mode, separate them with *, ×, or commas. The calculator processes them left to right.
What if a variable only appears in one monomial? Its exponent passes through unchanged. Multiplying x by y gives xy, and dividing 4x^2y by 2x leaves 2xy.
Are fractional coefficients allowed? Yes. Enter them as 1/2, 3/4, or as decimals like 0.5. The denominator of a coefficient fraction cannot be zero.
Why does the result sometimes show a negative exponent? Division can produce one when the denominator's exponent on a variable is larger than the numerator's. The answer is still correct, but it is no longer a standard polynomial monomial. Rewrite it as a fraction if needed.
Does the power input accept negative numbers? Yes, as long as it is a whole number. A negative power flips exponents in sign, and a zero power gives 1 for any nonzero coefficient.
