Dividing Monomials

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 ${\left({14}{{x}}^{{5}}\right)}\div{\left({7}{{x}}^{{2}}\right)}$.

Since fraction denotes division, then it is better (for visual perception) to rewrite ${\left({14}{{x}}^{{5}}\right)}\div{\left({7}{{x}}^{{2}}\right)}$ as $\frac{{{14}{{x}}^{{5}}}}{{{7}{{x}}^{{2}}}}$.

$\frac{{{14}{{x}}^{{5}}}}{{{7}{{x}}^{{2}}}}=$

$=\frac{{14}}{{7}}\cdot\frac{{{{x}}^{{5}}}}{{{{x}}^{{2}}}}=$ (multiplication of fractions in reverse direction: $\frac{{{a}\cdot{c}}}{{{b}\cdot{d}}}=\frac{{a}}{{b}}\cdot\frac{{c}}{{d}}$)

$={2}\frac{{{{x}}^{{5}}}}{{{{x}}^{{2}}}}=$ (simplify coefficient)

$={2}{{x}}^{{{5}-{2}}}=$ (rule for subtracting exponents)

$={2}{{x}}^{{3}}$ (simplify).

Answer: ${\left({14}{{x}}^{{5}}\right)}{\left({7}{{x}}^{{2}}\right)}={2}{{x}}^{{3}}$.

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 $\frac{{\frac{{1}}{{2}}{{x}}^{{4}}{{y}}^{{5}}{z}}}{{-{7}{{z}}^{{3}}{x}{{y}}^{{3}}}}$.

$\frac{{\frac{{1}}{{2}}{{x}}^{{4}}{{y}}^{{5}}{z}}}{{-{7}{{z}}^{{3}}{x}{{y}}^{{3}}}}=$

$=\frac{{\frac{{1}}{{2}}}}{{-{7}}}\cdot\frac{{{{x}}^{{4}}}}{{{x}}}\cdot\frac{{{{y}}^{{5}}}}{{{{y}}^{{3}}}}\cdot{\left(\frac{{z}}{{{z}}^{{3}}}\right)}$ (break down fraction)

$=-\frac{{1}}{{14}}\cdot{{x}}^{{{4}-{1}}}\cdot{{y}}^{{{5}-{2}}}\cdot{{z}}^{{{1}-{3}}}=$ (rule for subtracting exponents).

$=-\frac{{1}}{{14}}{{x}}^{{3}}{{y}}^{{3}}{{z}}^{{-{2}}}$ (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.

$=-\frac{{1}}{{14}}{{x}}^{{3}}{{y}}^{{3}}\frac{{1}}{{{{z}}^{{2}}}}=$ (get rid of negative exponent)

$=-\frac{{{{x}}^{{3}}{{y}}^{{3}}}}{{{14}{{z}}^{{2}}}}$ (write more compactly, using rule for multiplying fractions)

Answer: ${\left(\frac{{1}}{{2}}{{x}}^{{4}}{{y}}^{{5}}{z}\right)}\div{\left(-{7}{{z}}^{{3}}{x}{{y}}^{{3}}\right)}=-\frac{{{{x}}^{{3}}{{y}}^{{3}}}}{{{14}{{z}}^{{2}}}}$.

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 ${\left({4}{{x}}^{{3}}{{y}}^{{2}}\right)}\div{\left({3}{{z}}^{{5}}{y}{{x}}^{{3}}\right)}\div{\left(-{2}{{y}}^{{2}}{z}{x}\right)}$.

You need to do it step by step.

First divide ${\left({4}{{x}}^{{3}}{{y}}^{{2}}\right)}\div{\left({3}{{z}}^{{5}}{y}{{x}}^{{3}}\right)}$: $\frac{{{4}{{x}}^{{3}}{{y}}^{{2}}}}{{{3}{{z}}^{{5}}{y}{{x}}^{{3}}}}=\frac{{{4}{y}}}{{{3}{{z}}^{{5}}}}$.

Now, divide resulting fraction by third monomial (using rule for dividing fractions): $\frac{{{4}{y}}}{{{3}{{z}}^{{5}}}}\div{\left(-{2}{{y}}^{{2}}{z}{x}\right)}=\frac{{\frac{{{4}{y}}}{{{3}{{z}}^{{5}}}}}}{{-{2}{{y}}^{{2}}{z}{x}}}=\frac{{{4}{y}}}{{{3}{{z}}^{{5}}\cdot{\left(-{2}{{y}}^{{2}}{z}{x}\right)}}}=\frac{{2}}{{{3}{x}{y}{{z}}^{{6}}}}$.

Answer: ${\left({4}{{x}}^{{3}}{{y}}^{{2}}\right)}\div{\left({3}{{z}}^{{5}}{y}{{x}}^{{3}}\right)}\div{\left(-{2}{{y}}^{{2}}{z}{x}\right)}=\frac{{2}}{{{3}{x}{y}{{z}}^{{6}}}}$.

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 $\frac{{{7}{{a}}^{{7}}}}{{{3}{{a}}^{{2}}}}$.

Answer: $\frac{{7}}{{3}}{{a}}^{{5}}$.

Exercise 2. Simplify ${\left(-\frac{{1}}{{3}}{{x}}^{{3}}{{y}}^{{2}}{{z}}^{{7}}\right)}\div{\left(-\frac{{1}}{{6}}{{y}}^{{7}}{{z}}^{{3}}{{x}}^{{4}}\right)}$.

Answer: $\frac{{{2}{{z}}^{{4}}}}{{{x}{{y}}^{{5}}}}$.

Exercise 3. Divide the following: ${\left(-\frac{{3}}{{5}}{{c}}^{{2}}{{a}}^{{3}}{{b}}^{{7}}\right)}\div{\left(-\frac{{2}}{{7}}{{b}}^{{4}}{a}{{c}}^{{3}}\right)}\div{\left(-{a}{b}{c}\right)}$.

Answer: $-\frac{{{21}{a}{{b}}^{{2}}}}{{{10}{{c}}^{{2}}}}$.