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I'm given the complex rational expression:

$\frac{\frac{a}{b} - \frac{b}{a}}{\frac{a+b}{ab}}$

And asked to simplify. The solution provided is $a - b$ however I get $\frac{a^2 - b^2}{a+b}$

My working: Numerator first:

$\frac{a}{b} - \frac{b}{a}$ Least common denominator is $ab$

Multiplying each part to get the lcd in the denominator on both sides I get:

$\frac{a}{b} * \frac{a}{a}$ - $\frac{b}{a} * \frac{b}{b}$

= $\frac{a^2 - b^2}{ab}$

Multiplying this expression by the reciprocal in the original problem:

$\frac{a^2 - b^2}{ab}$ * $\frac{ab}{a+b}$

= $\frac{ab(a^2-b^2)}{ab(a+b)}$

Cancel out common factor ab:

$\frac{a^2 - b^2}{a + b}$

Where did I go wrong and how can I arrive at $a - b$?

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  • 10
    $\begingroup$ Soooo close. Factor the top. $\endgroup$ – Randall Feb 19 at 16:46
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Not that $$\frac{a}{b}-\frac{b}{a}=\frac{(a-b)(a+b)}{ab}$$ so we get $$\frac{(a-b)(a+b)ab}{(a+b)ab}=a-b$$

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$$a^2-b^2=(a-b)(a+b)$$ If $a+b\ne 0$, you can divide both the numerator and denominator by it.

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