Example QC #1
If only we had calculators…then this problem would be a piece of cake! Well, for those of you taking the new GRE, you will have a calculator, so this problem would be a breeze. But for those of you who are taking the current GRE, you’ll probably have to find some way to compare them without actually evaluating the radicals.
So let’s look at how we can change the numbers into a form that is easier to compare. The one on the left looks like it’s as simple as it’s going to get, so let’s work on the one on the right. We want to try to make it look like the one on the left so that it’s easier to compare.
Notice that there is a ½ on the outside of the radical (the square root sign). If there’s a number outside of the radical, we can always square it and then move it under the radical. Why? Well let’s look at a simple example:
√ 20 = √ 4×5
If we have two numbers being multiplied together underneath a radical, we can separate them, give them their own personal radicals, and then multiply those two radicals together. For instance:
√ 4×5 =√ 4 × √ 5
The square root of four is two, so
√ 4 × √ 5 = 2 × √ 5
Now, if we work backwards, we can see that we can move the 2 back into the radical by squaring it and multiplying it by the 5.
2 × √ 5 = √ 4×5 = √ 20
So now, back to our problem. Let’s move the ½ into the radical.
½√ 10 = √ ¼×10 = √ 10/4
The fraction 10/4 can be simplified to 5/2, and the radical can be rewritten as
√ 10/4 = √ 5/2
Hey! Look at that — it’s the exact same as the expression in column A. Thus, the answer is (C).
Example QC #2
This problem is fairly simple, but it does have a slightly tricky part to it that could make many people miss the question. I’m willing to bet that most people know the rule at hand here, but forgot it. Careless errors are the worst, so make sure you’re not missing anything!
You’re comparing x and y, so let’s solve for them.
√ 2x = 4 [square both sides]
2x = 16 [divide both sides by 2]
x = 8
So x is 8.
y2 = 64 [take the square root of both sides]
y = ±8
y is +8 or -8. Remember that when you take the square root of a number, it can be both positive and negative!
If y is +8, then they are the same; if y is -8, then x > y. Therefore, the answer is (D) because both possibilities are possible.
NOTE (3/21/2011): I should clarify that the square root symbol itself is by definition positive. What this means is:
√2 is a positive number; -√2 is a negative number.
√2 = ±1.4142
Only when you actually evaluate the square root (actually take the square root) is it both positive and negative. So if I write √2, I mean positive √2. If I evaluate √2, then I get ±1.4142. The square root symbol does not mean “positive and negative square root two.” In the above example, because we evaluated the square root (we actually took the square root of 64), it was ±8. If this is confusing, please let me know!