In this section we are going to explain how to divide monomials step by step. We have to differentiate between dividing two monomials, where we only have factors and when we have some sign of addition or subtraction in the middle, that is, when we have to divide polynomials of more than one term.
Division of two Monomials
As we already know, all the components of a monomial are multiplying each other, or in other words, the terms are made up of factors.
To divide two terms, we have to apply the properties of the powers, more specifically the property of power division, which indicates that when the powers have the same base, the base is maintained and its exponents are subtracted.
Example of How to Divide Monomials
Let’s see an example and solve it step by step:
1 – We start by dividing the numbers. To do this we can factor them previously or directly indicate the result:
Y we add it in the result:
2 – We continue with the variable x. We solve it separately to better follow the procedure. The base is maintained and the exponents are subtracted:
We add it to the result:
3 – Now we go with the variable y. We have the same exponent in the numerator and in the denominator and in this case, they are directly annulled. But be careful, the result of being annulled is 1, it is not 0.
This is so, because if we proceed in the same way, the result of dividing two equal powers is that the base is raised to 0 and therefore, any value raised to 0 equals 1.
In general, when the same factor is repeated in the numerator and denominator, the result is 1, which is the same as dividing any number, between itself.
Following our example, we add it to the result:
Normally, when the result is 1, nothing is indicated, but I write it to make it clearer.
3 – Finally, we do the same with the variable z:
We remember that negative exponents, in order to convert them into positive ones, are passed to the denominator (or to the numerator if it was already in the denominator). I dedicate an entire lesson to the minus signs in the powers in the course of powers.
4 – To finish we simplify by multiplying each term in the numerator and in the denominator:
I repeat again to make it clear that this procedure is valid only when we have factors.
If an addition or subtraction sign appears it would no longer be valid, because then it would no longer be a monomial, as there is more than one term and we would be talking about the division of polynomials, which is resolved differently, depending on whether it can be factored or we are asked to perform the division directly