Theory:

A product of numbers, variables and their exponents is called a monomial.
Example:
square.png
 
xx=x2
 
 
cube.png
 
aab=a2b
 
 
15x2.png
 
3x5x=(35)(xx)==15x2
Important!
Typically, we won't keep multiple signs between numerical factors and variables (6ay \(= 6ay\))
Important!
Note that one variable, such as \(x\), is also considered a monomial, because x=1x. A number like \(3\) is also a monomial, because 3=3x0
Some monomials can be simplified.
Example:
Let's simplify the monomial (6xy2(2)x3y) using the exponents' multiplication rule:         
aman=am+n
 
(6xy2(2)x3y) \(=\)6(2)xx3y2y=12x4y3

(numerical factors are multiplied, but the exponents of the variables are added)
Standard form of a monomial 
A monomial is in a standard form, if first there is a constant (numerical) factor, and then variable factors in alphabetical order. If there are several variable factors with the same base, you multiply them, thus you get one factor for each variable.
A monomial is in standard form, if:
  • every product of the same variables is written as one variable with an exponent aman=am+n;
  • a constant factor (or the monomial coefficient) is written as the first term of a monomial.
Example:
The standard form of the monomial (6xy2(2)x3y) is 12x4y3
A constant factor of a standard form of a monomial is called the monomial coefficient.
Example:
The coefficient of the monomial 12x4y3 is \(-12\).
Coefficients \(1\) and \(-1\) are not usually written.
 
1a2y=a2y
 
1x3=x3
Degree of a monomial is the sum of all the exponents of its variable terms (or factors).
Example:
12x4y3 is a seventh degree monomial, because \(4 + 3 = 7\).
Example:
monomial1.png