### Division of Monomials

According to Study.com:
"Monomials are the product of a coefficient, and a variable or variables."

from Cue Math teaching website:
"Practice Questions on Dividing Monomials"
"Q.1. Divide. 15a^2b^3 ÷ 5b"

The correct answer is listed as: 3a^2b^2

That means that even though an obelus was used to indicate "divided by," the statement was treated as a top-and-bottom vertical fraction, with 15a^2b^3 as the numerator & 5b as the denominator -- with NO PARENTHESES anywhere in the statement.

The "2" in "2x" is not a stand-alone number -- it's the coefficient in a monomial which tells you how many times to multiply a quantity (which is actually adding the quantity to itself), in much the same way that an exponent tells you how many times to multiply the base number by itself. Just as the exponent is "attached" to the base number, the coefficient of a monomial is "attached" to the variable (factor). Thus, the monomial division statement "2x divided by 2x" is: [ x + x ] ÷ [ x + x ]. Notice that the coefficient "disappears" when the statement is written out in its most basic form (as the indicated additions of the quantity). That proves, once and for all, that "peeling off" the coefficient of the monomial (the "2" in "2x") & using it in some other operation is not valid.

In the statement "6 ÷ 2(1+2)," the numerator "6" can be factored out as: "2(1+2)," making the statement:

2(1+2) ÷ 2(1+2)

Replacing what's inside the parentheses with the variable "x," the monomial division is:

2x ÷ 2x

If x equals 3 [expressed as (1+2) ], then the statement "2x ÷ 2x" can also be written as: "6 ÷ 2(1+2)," which is 6 divided by 6 -- which has a quotient of 1.

Division is fractions. Fractions is division. Do all the operations indicated in the numerator, then do all the operations indicated in the denominator, and finally divide the numerator by the denominator. Division has to go LAST. PEMDAS is incorrect for division statements -- because they're fractions.