Division of algebraic expressions

Our first examples of division of algebraic expressions involve simplifying and canceling.

Example 1

Simplify $\displaystyle\frac{{{3}{a}{b}{\left({4}{a}^{2}{b}^{5}\right)}}}{{{8}{a}^{2}{b}^{3}}}$

Show answer

First, we multiply out the top line:

$\displaystyle\frac{{{12}{a}^{3}{b}^{6}}}{{{8}{a}^{2}{b}^{3}}}$

When we write it out in full, this means

$\displaystyle\frac{{{12}\times{a}{a}{a}\times{b}{b}{b}{b}{b}{b}}}{{{8}\times{a}{a}\times{b}{b}{b}}}$

Next, cancel the numbers top and bottom (we divide top and bottom by $\displaystyle{4}$), the "a" terms (we cancel $\displaystyle{a}^{2}={a}{a}$from top and bottom) and the "b" terms (we cancel $\displaystyle{b}^{3}={b}{b}{b}$ from top and bottom) to give us the final answer:

$\displaystyle\frac{{{3}{a}{b}^{3}}}{{2}}$

Example 2

Simplify $\displaystyle\frac{{{12}{m}^{2}{n}^{3}}}{{{\left({6}{m}^{4}{n}^{5}\right)}^{2}}}$

Show answer

We square the denominator (bottom) of the fraction:

$\displaystyle\frac{{{12}{m}^{2}{n}^{3}}}{{{\left({6}{m}^{4}{n}^{5}\right)}^{2}}}=\frac{{{12}{m}^{2}{n}^{3}}}{{{36}{m}^{8}{n}^{10}}}$=36m8n1012m2n3​

Next, we cancel out the numbers, and the "m" and "n" terms to give the final answer:

$\displaystyle\frac{1}{{{3}{m}^{6}{n}^{7}}}$

Example 3

Simplify $\displaystyle\frac{{{6}{p}^{3}{q}^{2}-{10}{p}^{2}{q}}}{{{4}{q}}}$

Show answer

With this example, we'll break it into 2 fractions, both with denominator 4q to make it easier to see what to do.

$\displaystyle\frac{{{6}{p}^{3}{q}^{2}-{10}{p}^{2}{q}}}{{{4}{q}}}=\frac{{{6}{p}^{3}{q}^{2}}}{{{4}{q}}}-\frac{{{10}{p}^{2}{q}}}{{{4}{q}}}$=4q6p3q2​−4q10p2q​

Next, we cancel the numbers and variables:

$\displaystyle\frac{{{3}{p}^{3}{q}}}{{2}}-\frac{{{5}{p}^{2}}}{{2}}$−25p2​

Finally, we combine the fractions:

$\displaystyle\frac{{{3}{p}^{3}{q}-{5}{p}^{2}}}{{2}}$

After you have had some practice with these, you'll be able to do it without separating them into 2 fractions first.

Share on Google Plus

About Unknown

This is a short description in the author block about the author. You edit it by entering text in the "Biographical Info" field in the user admin panel.