What is a monomial?

To be able to understand what a monomial is, we first need to know about the following terms:

1. Constants and variables: A symbol in algebra having a fixed value is called a constant, whereas, a symbol which can be assigned different values is better known as a variable.
For example, 1/3,-8,¶,√2 etc are all constants and x,y,z etc are all variables.

2. Algebraic expression could be explained as a combination of constants along with the variables and connected by the signs +,-,x,÷ is called an algebraic expression. The several parts of an expression separated by + or – sign are called the terms of the expression. Thus 5 - 3x + 4y + 8xy +7x^2y is an algebraic expression containing five terms and 5x – 7y is an algebraic expression containing two terms.

Monomial definition: A monomial is an algebraic expression that contains only one term. For example: 5x, 7y, -8xy, 4x^2yz^3 etc are all monomials. Monomials would not have any + or – signs between constants or variables. 2x + 5, 3x - 7y, x^2 + 2xy + y^2 etc are not monomials because they have more than one term. They are called polynomials.

Multiplying  monomials:
Product of monomials = (product of numerical coefficients) x (product of literal factors)

Example 1: Multiply 16x^2y^3 by 4x^4y^3z
Solution:
Product of numerical coefficients = 16*4 = 64
Product of literal factors = x^2y^3 * x^4y^3z = x^(2+4) * y^(3+3) * z = x^6y^6z
(The rule is: x^m * x^n = x^(m+n): add the exponents)
Therefore product of the monomials = 64x^6y^6z

Dividing monomials:
Quotient of two monomials = (Quotient of their numerical coefficients) x (Quotient of their literal factors)

Example 2: Divide 32a^3b^3 by -8ab
Solution:
Quotient of numerical coefficients = 32/(-8) = -4
Quotient of literal factors = a^3b^3/ab = (a^3/a) * (b^3/b) = a^(3-1) * b^(3-1) = a^2b^2
(The rule is: x^m ÷ x^n = x^(m-n): subtract the exponents)
Therefore the quotient of the monomials = -4a^2b^2

Example 3: Simplify -81a^5b^4c^3 * 3abc ÷ 9a^2b^2c^2
Solution:
First the multiplication of the first two monomials = -81*3 * a^5b^4c^3 * abc
= -243a^6b^5c^4.
Now the division: -243a^6b^5c^4 ÷ 9a^2b^2c^2 = -27a^4b^3c^2