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Testing Numbers in Quantitative Comparison

\(x\), \(p\), and \(q\) are integers, and \(p\) is greater than \(q\)

Quantity A                              Quantity B

 \( x^{2p}\)                                             \(x^{2q}\)

(A) Quantity A is greater.

(B) Quantity B is greater.

(C) The two quantities are equal.

(D) The relationship cannot be determined from the information given.

Inequalities are tricky. Whenever you see one in a QC question you should proceed with caution.

Exponents are also a trouble spot for most students, not because they are inherently tricky, like inequalities, but because our intuitions about them are often misleading.

In this particular problem we’re dealing with integers, so the realm of possible values for [math]x[/math], [math]p[/math], and [math]q[/math] is pretty restricted. Granted, there are an infinite number of integers, but we like integers. They’re “easier” than fractions, decimals, and irrational numbers like [math]\sqrt{2}[/math].

We have a further restriction – the exponents are even because they are both multiples of two. Now we don’t have to worry about [math]x[/math] being negative because any integer to an even power will always be positive.

Obviously, larger exponents produce larger quantities, so the answer must be (A). Right?

WRONG!

The reasoning above is reasonably solid, but it’s missing an essential detail. If you guessed (A) take a second and see if you can find an exception to the “rule” that  larger exponents produce larger quantities.

Did you find the exception? There are actually three exceptions in this case.

What if \(x\) is 0, 1 or -1? Then the two quantities would be equal.

Students often forget to check these numbers. But it’s an essential part of picking numbers to solve QC problems because 0, 1, and -1 are oddballs, and oddballs produce exceptions.

So, the answer is (D)

When I pick numbers I use the acronym P0FN1 to make sure I’m not missing something.

P – Postitves

0 – Zero

F – Fractions

N – Negatives

1 – One