The complex fractions just keep coming, and some of them come with surprises. Let's dig in!

- Simplify the following complex fraction:

Can I start my simplification by hacking off the *x* − 3's? Can I cancel the 4 with the 12? Or any of the 3's with the 9 or the 12? (Hint: Really, no!)

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The common denominator for this complex fraction is *x* − 3, so I'll multiply through, top and bottom, by that.

Clearly, nothing else cancels, so my final answer is:

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It is highly unusual for a complex fraction to simplify this much, but it can happen. In this case, the "except for *x* equal to 3" part is rather important, since the original fraction is not always equal to ^{3}/_{4}. Indeed, it is not even defined for *x* equal to 3 (since this would cause division by zero). If the original expression had been set equal to *y* and the instructions on the exercise had been to graph the function, the result would have been the graph of a straight line with a hole; the line would have been horizontal at a height of three-fourths of a unit above the *x*-axis, with an open hole at *x* = 3.

By the way, those with sharp eyes may notice something about the above exercise that I'd missed; namely, it was possible to factor out in the first step, and cancel to get the answer right away. This would have looked like the following:

The only downside of the above cleverness is that it is harder to locate the forbidden values of the variable, because there was no step at which an *x* was cancelled off. While the one restriction (namely, *x* ≠ 3, from the denominators in the smaller fractions), the second restriction is not obvious. In such a case, one has to find the value(s) of the variable which would cause the denominator of the entire "stacked" fraction to be zero (because this would cause division by zero for the stacked fraction). In the above case, this process would be the following:

Yes, complex fractions can be messy and involved.

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- Simplify the following complex rational expression:

Can I start my simplification by canceling multiplications with additions, or between the numerator of one fraction and the denominator of another fraction to which it is added (rather than multiplied), as shown below?

**DON'T DO THIS!**

**NOT EVER!**

I can only cancel factors within the same fraction, not terms or portions of different added fractions, so the above cancellations are not in any way proper.

Instead, the first thing I will do is multiply through, top and bottom, by the common denominator of *xy*.

Then my final answer is:

*y* − *x*, for *x*, *y* ≠ 0, *x* ≠ −*y*

- Simplify the following expression:

Can I start by canceling off the 1's or the 1/*t*'s? (Hint: Still really no!)

I first note that the variable cannot equal zero, as this would cause division by zero. Moving on, I multiply through, top and bottom, by the common denominator of *t*.

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Can I reach inside the "understood" parentheses that enclose the polynomial expressions in the numerator and denominator, and cancel off just the *t*'s now? Or cancel off just the 1's? (Hint: Really, really, no!)

I can only cancel factors, not terms inside larger expressions. At this point, there is nothing left that factors, so this is as simplified as it gets. Then my final hand-in answer is:

When working with complex fractions, be careful to show each step completely. Don't try to skip steps or do everything in your head. And don't get careless with cancellation; remember that you can only cancel factors, not terms that are parts of larger expressions. If you keep these things straight and do your work clearly, you should be fairly successful with these problems. Just don't rush; give yourself time for familiarity to grow.

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