# Dividing Monomials

## Related Calculator: Polynomial Calculator

**Monomials can be divided** in the same manner as numbers.

To divide monomials, we use properties of fractions and properties of exponents.

Let's start from a simple example, involving only one variable.

**Example 1**. Simplify `(14x^5)-:(7x^2)`.

Since fraction denotes division, then it is better (for visual perception) to rewrite `(14x^5)-:(7x^2)` as `(14x^5)/(7x^2)`.

`(14x^5)/(7x^2)=`

`=14/7*(x^5)/(x^2)=` (multiplication of fractions in reverse direction: `(a*c)/(b*d)=a/b*c/d`)

`=2(x^5)/(x^2)=` (simplify coefficient)

`=2x^(5-2)=` (rule for subtracting exponents)

`=2x^3` (simplify).

**Answer**: `(14x^5)(7x^2)=2x^3`.

Let's see how to multiply, if there are more than one variable (in fact technique is same).

Also, following examples, shows how to handle negative exponents.

**Example 2**. Simplify `(1/2x^4y^5z)/(-7z^3xy^3)`.

`(1/2x^4y^5z)/(-7z^3xy^3)=`

`=(1/2)/(-7)*(x^4)/(x)*(y^5)/(y^3)*(z/z^3)` (break down fraction)

`= -1/14*x^(4-1)*y^(5-2)*z^(1-3)=` (rule for subtracting exponents).

`= -1/14x^3y^3z^(-2)` (simplify).

You can leave answer as it is, but, in most cases, teachers don't allow negative exponents, so you need to get rid of negative exponents.

`= -1/14x^3y^3 1/(z^2)=` (get rid of negative exponent)

`= -(x^3y^3)/(14z^2)` (write more compactly, using rule for multiplying fractions)

**Answer**: `(1/2x^4y^5z)-:(-7z^3xy^3)=-(x^3y^3)/(14z^2)`.

**Note**: last example shows, that result of division monomial by monomial is not always monomial.

We can even divide more than two monomials!

**Example 3**. Divide `(4x^3y^2)-:(3z^5yx^3)-:(-2y^2zx)`.

You need to do it step by step.

First divide `(4x^3y^2)-:(3z^5yx^3)`: `(4x^3y^2)/(3z^5yx^3)=(4y)/(3z^5)`.

Now, divide resulting fraction by third monomial (using rule for dividing fractions): `(4y)/(3z^5)-:(-2y^2zx)=((4y)/(3z^5))/(-2y^2zx)=(4y)/(3z^5*(-2y^2zx))=2/(3xyz^6)`.

**Answer**: `(4x^3y^2)-:(3z^5yx^3)-:(-2y^2zx)=2/(3xyz^6)`.

**Note**: after some practice, you will want to skip some steps and just divide "like" variables immediately, making commutations and splittings in your head.

Now, it is time to exercise.

**Exercise 1**. Divide `(7a^7)/(3a^2)`.

**Answer**: `7/3a^5`.

**Exercise 2**. Simplify `(-1/3x^3y^2z^7)-:(-1/6y^7z^3x^4)`.

**Answer**: `(2z^4)/(xy^5)`.

**Exercise 3**. Divide the following: `(-3/5c^2a^3b^7)-:(-2/7b^4ac^3)-:(-abc)`.

**Answer**: `-(21ab^2)/(10c^2)`.