Let’s look the product of two monomials [[$ 3a^2 $]] and [[$ 5a^4 $]] Both terms are formed from a coefficient and a product of a variable. As a result, the product calculation [[$ 3a^2 \cdot 5a^4 $]] can be simplified as follows:

Multiplying a monomial by a monomial

Multiply the coefficients of the terms by one another. Multiply the variables of the terms by one another.

If monomials have the same letters as variables, the rules for calculating powers are applied to their multiplication. If the variables have different letters, they remain in multiplication form and cannot be combined into powers.

Examples

Example 1

Simplify the expressions.

a) [[$ 4 \cdot 3x = 12x $]]
b) [[$ 2 \cdot (-4y) = 2 \cdot (-4) \cdot y = -8y $]]
c) [[$ -5x^2 \cdot (-3) = -5 \cdot (-3) \cdot x^2 = 15x^2 $]]
d) [[$ 3 \cdot (-xy) = 3 \cdot (-1) \cdot xy = -3xy $]]

Example 2

Calculate the products of the following monomials.

a) [[$ 2x \cdot (-3x) = 2 \cdot (-3) \cdot x \cdot x = -6x^2 $]]
b) [[$ 4a \cdot 5b = 4 \cdot 5 \cdot a \cdot b = 20ab $]]
c) [[$ -x \cdot (-3y) = (-1) \cdot (-3) \cdot x \cdot y = 3xy $]]
d) [[$ 2y \cdot 4y^2 \cdot (-y^2) = 2 \cdot 4 \cdot (-1) \cdot y \cdot y^2 \cdot y^2 =-8 \cdot y^{(1+2+2)} = -8y^5 $]]
e) [[$ -5ab \cdot 2a = -5 \cdot 2 \cdot a \cdot a \cdot b = -10a^2b $]]