Hi Mike! Is there a “best” way to do #25 on dSAT 3 module 1? Thanks!

The first step on any question like this, where you’re presented with just a spaghetti pile of x‘s and y‘s is to stop, breathe, and look for a foothold. You know what test you’re taking, and you know that there is supposed to be a way to get through this question relatively quickly, say in a minute (or two, for the hardest ones at the end of a section, like this one).

To add fractional expressions, which we have to do here, what do we always need? A common denominator! So look at the denominators in the given expression. Both have an x-8 structure; that’s our foothold. So let’s factor the more complicated denominator on the right.

x^2y-8xy=(x-8)(xy)

So to add these expressions, we’ll need to multiply the left-hand side by a useful form of one: \dfrac{xy}{xy}=1.

At this point we could just grind it out, but the “best” way to do this is to keep one eye on the answer choices while we go. And on this particular question, that will save us a ton of time.

Remember, when we’re adding fractional expressions, once we have found a common denominator, the denominator doesn’t change. In other words:

\dfrac{\text{complicated numerator}}{\text{common denominator}}+\dfrac{\text{different complicated numerator}}{\text{common denominator}}\\\\\\=\dfrac{\text{very complicated numerator}}{\text{same common denominator}}

In this case, we figured out a few paragraphs ago that we can use x^2y-8xy as our common denominator. So that should be the denominator in our result. AND IT ONLY SHOWS UP IN ONE ANSWER CHOICE!

Do as much algebraic manipulation as you want to make sure you’re comfortable, but for me, I’m just doing a quick scan of the other denominators to make sure they’re not equivalent, and then I’m picking choice C and moving on.

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