Master monomial operations with step-by-step explanations. Compute product, quotient, degree, numeric evaluation, and simplified form.
A monomial is an algebraic expression consisting of exactly one term. It includes a numerical coefficient (real number) multiplied by one or more variables raised to non‑negative integer exponents. Classic examples: 7x²y³, -3ab²c, 12 (constant monomial). Monomials are the building blocks of polynomials and play a fundamental role in algebra, calculus, and applied mathematics.
General form: \( a \cdot x_1^{n_1} x_2^{n_2} \dots x_k^{n_k} \) where \(a \in \mathbb{R}\) and \(n_i \in \mathbb{Z}_{\ge 0}\).
When multiplying monomials, we apply the product rule: \(x^m \cdot x^n = x^{m+n}\). For division: \(x^m / x^n = x^{m-n}\) (provided \(x \neq 0\)). Raising a monomial to a power: \((a x^m y^n)^k = a^k x^{mk} y^{nk}\). These rules ensure that the result of operating on monomials is another monomial (unless negative exponents appear, which yield rational expressions).
| Operation | Rule | Example |
|---|---|---|
| Multiplication | Multiply coefficients, add exponents of each variable | (3x²y)(2xz) = 6x³yz |
| Division | Divide coefficients, subtract exponents | 8x⁴y³ ÷ 2x²y = 4x²y² |
| Power | Raise coefficient to power, multiply each exponent | (2x²y³)² = 4x⁴y⁶ |
Our monomial calculator uses precise algebraic rules:
We also compute the total degree (sum of variable exponents) and provide a human-readable algebraic form with superscripts.
Engineers use monomials to model physical quantities: the volume of a rectangular box with side lengths \(2x\), \(3x^2\), \(y\) is \(6x^3y\) — a monomial. Multiplying monomials streamlines dimensional analysis. In computer algebra systems, monomial simplification reduces computational complexity.