Table of Contents

Last modified on January 10th, 2024

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, 15p^{7} and 3p^{2}.

**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**

= 5p^{7-2}

Thus, the quotient is 5p^{5}.

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 6m^{1-2}n^{3} = 6m^{-1}n^{3}

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

**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 -3p^{1-1}q

Thus, the quotient is -3q

**Divide 25p ^{15}q^{13} by 5p^{24}q^{4}.**

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 16m ^{4}y^{2} and 24my^{2}?**

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 ${\dfrac{2}{3}}$