4.13:_Algebra_Connections

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<p class="lt-math-10591">In an advanced algebra course students are often asked to work with complicated expressions like:</p>

<p class="lt-math-10591">\[\frac{\frac{1}{x} + 1}{\frac{3}{x}} \nonumber \]</p>

<p class="lt-math-10591">We can make it look friendlier by using the key fraction rule, <u>exactly the same technique we used</u> in the chapter on &ldquo;Dividing Fractions: Invert and Multiply.&rdquo; In this example, let us multiply the numerator and denominator each by <img alt="x" height="8" title="Rendered by QuickLaTeX.com" width="10" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/16395/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.svg?revision=1" />. (Do you see why this is a good choice?) We obtain:</p>

<p class="lt-math-10591">\[\frac{(\frac{1}{x} + 1) \cdot x}{(\frac{1}{x}) \cdot x} = \frac{1 + x}{3}, \nonumber \]</p>

<p class="lt-math-10591">and \(\frac{1+x}{3}\) is much less scary.</p>

<p class="lt-math-10591">Notice that expressions like</p>

<p class="lt-math-10591">\[\frac{1}{x} \nonumber \]</p>

<p class="lt-math-10591">cannot be rewritten as a decimal. Expressions like this arise in numerous applications, so it is important for math and science students to be able to work with fractions in fraction form, without always resorting to converting to decimals.</p>

<section class="box-example"><span id="Example_.5C(.5CPageIndex.7B1.7D.5C):"></span><h5 class="box-legend lt-math-10591"><span class="lt-icon-default">Example \(\PageIndex{1}\):</span></h5>

<p class="lt-math-10591">As another example, given:</p>

<p class="lt-math-10591">\[\frac{\frac{1}{a} - \frac{1}{b}}{ab}, \nonumber \]</p>

<p class="lt-math-10591">one might find it helpful to multiply the numerator and the denominator each by <img alt="a" height="8" title="Rendered by QuickLaTeX.com" width="9" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/16625/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.svg?revision=1" /> and then each by <img alt="b" height="13" title="Rendered by QuickLaTeX.com" width="8" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/16392/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.svg?revision=1" />:</p>

<p class="lt-math-10591">\[\frac{(\frac{1}{a} - \frac{1}{b}) \cdot a \cdot b}{(ab) \cdot a \cdot b} = \frac{b - a}{a^{2} b^{2}} \ldotp \nonumber \]</p>
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<section class="box-example"><span id="Example_.5C(.5CPageIndex.7B2.7D.5C):"></span><h5 class="box-legend lt-math-10591"><span class="lt-icon-default">Example \(\PageIndex{2}\):</span></h5>

<p class="lt-math-10591">\[\frac{\frac{1}{(w+1)^{2}} - 2}{\frac{1}{w+1)^{2}} + 5}, \nonumber \]</p>

<p class="lt-math-10591">it might be good to multiply numerator and denominator each by \((w+1)^{2}\). (Why?)</p>

<p class="lt-math-10591">\[\frac{(\frac{1}{(w+1)^{2}} - 2) \cdot (w+1)^{2}}{(\frac{1}{w+1)^{2}} + 5) \cdot (w+1)^{2}} = \frac{1-2(w+1)^{2}}{1+5(w+1)^{2}} \ldotp \nonumber \]</p>
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<p class="lt-math-10591">Can you make each of these expressions look less scary?</p>

<p class="lt-math-10591">\[\frac{2 - \frac{1}{x}}{1 + \frac{1}{x}}, \qquad \frac{\frac{1}{x+h} + 3}{\frac{1}{x+h}}, \qquad \frac{1}{\frac{1}{a} + \frac{1}{b}}, \qquad \frac{\frac{1}{x+a} - \frac{1}{x}}{a} \ldotp \nonumber \]</p>

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