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When working with rational expressions, it is often best to write them in the simplest possible form. For example, the rational expression
$$\frac{{x}^{2}-4}{{x}^{2}-6x+8}$$
can be reduced to the simpler expression
$\frac{x+2}{x-4}$ for all
$x$ except
$x=2,4$ .
From our discussion of equality of fractions in Section
[link] , we know that
$\frac{a}{b}=\frac{c}{d}$ when
$ad=bc$ . This fact allows us to deduce that, if
$k\ne 0,\frac{ak}{bk}=\frac{a}{b},$ since
$akb=abk$ (recall the commutative property of multiplication). But this fact means that if a factor (in this case,
$k$ ) is common to both the numerator and denominator of a fraction, we may remove it without changing the value of the fraction.
$$\frac{ak}{bk}=\frac{a\overline{)k}}{b\overline{)k}}=\frac{a}{b}$$
The process of removing common factors is commonly called cancelling .
$\frac{16}{40}$ can be reduced to
$\frac{2}{5}$ . Process:
$\frac{16}{40}=\frac{2\xb72\xb72\xb72}{2\xb72\xb72\xb75}$
Remove the three factors of 1;
$\frac{2}{2}\xb7\frac{2}{2}\xb7\frac{2}{2}.$
$\frac{\overline{)2}\xb7\overline{)2}\xb7\overline{)2}\xb72}{\overline{)2}\xb7\overline{)2}\xb7\overline{)2}\xb75}=\frac{2}{5}$
Notice that in
$\frac{2}{5}$ , there is no factor common to the numerator and denominator.
$\frac{111}{148}$ can be reduced to
$\frac{3}{4}$ . Process:
$\frac{111}{148}=\frac{3\xb737}{4\xb737}$
Remove the factor of 1;
$\frac{37}{37}$ .
$\frac{3\xb7\overline{)37}}{4\xb7\overline{)37}}$
$\frac{3}{4}$
Notice that in
$\frac{3}{4}$ , there is no factor common to the numerator and denominator.
$\frac{3}{9}$ can be reduced to
$\frac{1}{3}$ . Process:
$\frac{3}{9}=\frac{3\xb71}{3\xb73}$
Remove the factor of 1;
$\frac{3}{3}$ .
$\frac{\overline{)3}\xb71}{\overline{)3}\xb73}=\frac{1}{3}$
Notice that in
$\frac{1}{3}$ there is no factor common to the numerator and denominator.
$\frac{5}{7}$ cannot be reduced since there are no factors common to the numerator and denominator.
Problems 1, 2, and 3 shown above could all be reduced. The process in each reduction included the following steps:
We know that we can divide both sides of an equation by the same nonzero number, but why should we be able to divide both the numerator and denominator of a fraction by the same nonzero number? The reason is that any nonzero number divided by itself is 1, and that if a number is multiplied by 1, it is left unchanged.
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