# Dividing Monomials

Dividing monomials is a way of simplifying that involves the division of their coefficients followed by that of their variables.

Here, we use the exponent rule ${\dfrac{a^{m}}{a^{n}}=a^{m-n}}$, where a ≠ 0, while dividing the variables with the same base and different exponents.

Let us divide the monomials, 15p7 and 3p2.

Step 1: Writing In Quotient Form

${\dfrac{15p^{7}}{3p^{2}}}$

Step 2: Dividing the Coefficients

= ${\left( \dfrac{15}{3}\right) \left( \dfrac{p^{7}}{p^{2}}\right)}$

= ${5\left( \dfrac{p^{7}}{p^{2}}\right)}$

Step 3: Dividing the Variables

= 5p7-2

Thus, the quotient is 5p5.

Let us consider another example ${\dfrac{18mn^{3}}{3m^{2}}}$

It is written as ${\left( \dfrac{18}{3}\right) \left( \dfrac{mn^{3}}{m^{2}}\right)}$

On dividing the coefficients, ${6\left( \dfrac{mn^{3}}{m^{2}}\right)}$

Now, applying the exponent rules, we get 6m1-2n3 = 6m-1n3

Thus, the quotient is ${6\dfrac{n^{3}}{m}}$.

## Solved Examples

Divide the monomials 15pq by -5p.

Solution:

On dividing 15pq by -5p, we get ${\dfrac{15pq}{-5p}}$
= ${-\left( \dfrac{15}{5}\right) \left( \dfrac{pq}{p}\right)}$
Applying the exponents rule, we get -3p1-1q
Thus, the quotient is -3q

Divide 25p15q13 by 5p24q4.

Solution:

On dividing the given monomials, we get ${\dfrac{25p^{15}q^{13}}{5p^{24}q^{4}}}$
= ${\left( \dfrac{25}{5}\right) \left( \dfrac{p^{15}q^{13}}{p^{24}q^{4}}\right)}$
Applying th exponents rule, we get ${5p^{15-24}q^{13-4}}$
= ${5p^{-9}q^{9}}$
= ${\dfrac{5q^{9}}{p^{9}}}$
Thus, the quotient is ${5\left( \dfrac{q}{p}\right) ^{9}}$

Simplify ${\dfrac{64a^{9}b^{11}c^{5}}{8a^{15}c^{2}}}$

Solution:

Here, ${\dfrac{64a^{9}b^{11}c^{5}}{8a^{15}c^{2}}}$
= ${\left( \dfrac{64}{8}\right) \left( \dfrac{a^{9}b^{11}c^{5}}{a^{15}c^{2}}\right)}$
= ${8\left( a^{9-15}b^{11}c^{5-2}\right)}$
= ${8\left( a^{-6}b^{11}c^{3}\right)}$
= ${\dfrac{8b^{11}c^{3}}{a^{6}}}$
Thus, ${\dfrac{64a^{9}b^{11}c^{5}}{8a^{15}c^{2}}}$ is simplified to ${\dfrac{8b^{11}c^{3}}{a^{6}}}$

What will be the coefficient of the quotient of 16m4y2 and 24my2?

Solution:

On dividing the given monomials, the quotient is written as ${\dfrac{16m^{4}y^{2}}{24my^{2}}}$
Now, by dividing the coefficients and the variables individually, we get
${\left( \dfrac{16}{24}\right) \left( \dfrac{m^{4}y^{2}}{my^{2}}\right)}$
Applying the exponents rule, we get
${\left( \dfrac{2}{3}\right) \left( m^{4-1}y^{2-2}\right)}$
= ${\left( \dfrac{2}{3}\right) m^{3}}$, which is the required quotient.
Thus, the coefficient of the quotient is ${\dfrac{2}{3}}$