GRE Answer Choice Tricks, Part 3

In my last entry, I showed you how to rejigger radical fractions to make them look the way the test makers want them to look. The example in that entry was a pretty easy one, though — all you had to do was multiply the top and bottom of the fraction by a radical. This time we’re going to crank it up a notch.

Imagine you worked out the math to a GRE quant problem and got this:

There’s a radical in the bottom of that fraction, so it will never be the credited answer to a GRE quantitative problem. As we saw last time, you’ve got to get the radical out from underneath the fraction. Some of you might perhaps be tempted to multiply the fraction through by the square root of 5, similar to what we did last time. Notice, though, what’ll happen to the bottom of the fraction if you do that:

Whoops! We got rid of the radical all right, but another one popped up in its place. What you have to do instead is multiply the top and bottom of the fraction by the complement of the denominator: that is, copy the denominator, but flip the sign. In this case, there’s a + to begin with, so switch it to a – and multiply by . Why do this? Well, you’re about to see!

Since this one’s more complicated, let’s do the two operations separately. First, on top, we’ve got:

Easy enough. The bottom is this:

And here’s why multiplying by the complement is so effective: this is a difference of squares. You absolutely have to know this property on Test Day:

So,

Notice how this gets rid of the radicals cleanly, without making any new ones! All together, then:

Finally, every term in that fraction is a multiple of 4, so divide through by 4 to get the answer:

Click, confirm, and another seemingly impossible GRE challenge bites it. Got any questions or points of confusion about manipulating radicals? Let us know in the comments!