# Why is $\frac{\frac{a}{b} - \frac{b}{a}}{\frac{a+b}{ab}}$ = a - b?

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 $$a - b$$?

• Soooo close. Factor the top. – Randall Feb 19 at 16:46

Not that $$\frac{a}{b}-\frac{b}{a}=\frac{(a-b)(a+b)}{ab}$$ so we get $$\frac{(a-b)(a+b)ab}{(a+b)ab}=a-b$$
$$a^2-b^2=(a-b)(a+b)$$ If $$a+b\ne 0$$, you can divide both the numerator and denominator by it.