algebra for division

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

Example 1

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

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

We square the denominator (bottom) of the fraction:

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

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

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

Easy to understand math videos:
MathTutorDVD.com

Example 3

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

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

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

Next, we cancel the numbers and variables:

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

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.

Continues below ⇩

Dividing by a Fraction

Recall the following when dividing algebraic expressions.

The reciprocal of a number x, is $\displaystyle\frac{1}{{x}}$.

For example, the reciprocal of 5 is $\displaystyle\frac{1}{{5}}$ and the reciprocal of $\displaystyle{1}\frac{2}{{3}}$ is $\displaystyle\frac{3}{{5}}$.

To divide by a fraction, you multiply by the reciprocal of the fraction.

For example, $\displaystyle\frac{3}{{4}}\div\frac{7}{{x}}=\frac{3}{{4}}\times\frac{x}{{7}}=\frac{{{3}{x}}}{{28}}$

Example 4

Simplify

$\displaystyle\frac{{{3}+\frac{1}{{x}}}}{{\frac{5}{{x}}+{4}}}$

I'll show you how to do this two different ways. It is worth seeing both, because they are both useful. You can decide which is easier ;-)

Solution 1 - Multiplying by the Reciprocal

I take the top expression (numerator) and turn it into a single fraction with denominator x.

$\displaystyle{3}+\frac{1}{{x}}=\frac{{{3}{x}+{1}}}{{x}}$

We do likewise with the bottom expression (denominator):

$\displaystyle\frac{5}{{x}}+{4}=\frac{{{5}+{4}{x}}}{{x}}$

So the question has become:

$\displaystyle\frac{{{3}+\frac{1}{{x}}}}{{\frac{5}{{x}}+{4}}}=\frac{{\frac{{{3}{x}+{1}}}{{x}}}}{{\frac{{{5}+{4}{x}}}{{x}}}}$

We think of the right side as a division of the top by the bottom:

$\displaystyle\frac{{{3}{x}+{1}}}{{x}}\div\frac{{{5}+{4}{x}}}{{x}}$

To divide by a fraction, you multiply by the reciprocal:

$\displaystyle\frac{{{3}{x}+{1}}}{{{x}}}\times\frac{x}{{{5}+{4}{x}}}=\frac{{{3}{x}+{1}}}{{{5}+{4}{x}}}$

The x's cancelled out, and we have our final answer, which cannot be simplified any more.

Solution 2 - Multiplying Top and Bottom

I recognise that I have "/x" in both the numerator and denominator. So if I just multiply top and bottom by x, it will simplify everything by removing the fractions on top and bottom.

$\displaystyle\frac{{{3}+\frac{1}{{x}}}}{{\frac{5}{{x}}+{4}}}\times\frac{x}{{x}}$

I am really just multiplying by "1" and not changing the original value of the fraction - just changing its form.
So I multiply each element of the top by x and each element of the bottom by x and I get:

$\displaystyle\frac{{{3}+\frac{1}{{x}}}}{{\frac{5}{{x}}+{4}}}\times\frac{x}{{x}}=\frac{{{3}{x}+{1}}}{{{5}+{4}{x}}}$

I cannot simplify any further.