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If a rectangle has a width of 3x3x and a length of 5x25x^2, its area is found by multiplying them together. But how do you multiply terms made of numbers and variables packed together?
A monomial is a single algebraic term made of numbers and variables multiplied together, such as −4x3-4x^3. The numerical factor is called the coefficient, and the raised number is the exponent showing how many times the base variable is multiplied by itself.
Before combining full monomials, what actually happens when variables with exponents collide?

The Product Rule for Exponents

When you multiply terms with the same base, you add their exponents: xa⋅xb=xa+bx^a \cdot x^b = x^{a+b}. This works because exponents simply count factors: x2⋅x3=(x⋅x)⋅(x⋅x⋅x)x^2 \cdot x^3 = (x \cdot x) \cdot (x \cdot x \cdot x), giving 55 total factors of xx.
Now that we know how powers combine, how do we handle both numbers and variables at the same time?

Multiplying Two Monomials

To multiply monomials, apply the Commutative and Associative Properties of Multiplication to regroup numerical coefficients together and like variable bases together: (axm)(bxn)=(a⋅b)(xm⋅xn)(a x^m)(b x^n) = (a \cdot b)(x^m \cdot x^n).
Let's multiply (4x3)(2x5)(4x^3)(2x^5). Regrouping gives (4⋅2)⋅(x3⋅x5)=8x3+5=8x8(4 \cdot 2) \cdot (x^3 \cdot x^5) = 8x^{3+5} = 8x^8.
What happens if a monomial has multiple different variables, or a variable without a visible exponent?