7.4.2 Simplification
This subsection is not completed yet, but in the meantime here are some practice questions.
Remember:
You can cancel common factors from the top and bottom of fractions: \[\frac{ab}{ac} = \frac{b}{c}\]
What we’re really doing is dividing the top and bottom by $a$.
Note that this only works for common factors. You can’t cancel common terms:
\[\frac{a + b}{a + c}\ne \frac{b}{c}\]
Remember:
If after cancelling there’s nothing left in the numerator, it becomes $1$: \[\frac{a}{ab} = \frac{1}{b}\]
because when we divide $a$ by $a$, we get $1$.
A common mistake is to make the numerator disappear, but this is wrong:
\[\frac{a}{ab} \ne b\]
Remember:
Subtract the smaller exponent from the larger exponent when simplifying fractions:
\begin{align*}
\frac{a^m b}{a^n c} &= \frac{a^{m-n} b}{c}\\
\frac{a^n b}{a^m c} &= \frac{b}{a^{m-n}c}
\end{align*}
if $m\gt n$. We’re dividing both top and bottom of the fraction by $a^n$.
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The How and Why of Mathematics by J L Gorham is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.