Workbook Wednesdays: X to the Nth Power Answer

Check out the answer to yesterday’s Challenge Problem!

Question
If n is a positive integer and x does not equal zero, is x^n > x^(n+1)?

1) x < 1

2) n is even.

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Solution
The answer to the question depends on the values of both x and n. Specifically, we care about the value of x^n, since this will determine how we can rephrase the question.

If n is even, then x^n > 0, no matter what the value of x is (remember that x is nonzero).
Likewise, if x > 0, then x^n > 0, no matter what the value of n is.

The reason that we care about the value of x^n is that we can simplify the question by dividing by x^n:

After we divide both sides of the inequality by x^n, the question “Is x^n > x^(n+1)?” becomes “Is 1 > x?” ONLY IF x^n > 0, which is true if x > 0 OR if n is even. (Recall that x is nonzero; thus, we are allowed to divide by x^n.) On the other hand, if x^n < 0, then the question rephrases to “Is 1 < x?”

Statement 1: INSUFFICIENT. We know that x < 1, but x could be positive or negative. Moreover, we do not know whether n is even or odd. As a result, we do not know the sign of x^n, and thus we do not know the answer to either the rephrased question or to the original question.

Alternatively, you can choose positive and negative values of x and an odd n, in order to test the question. If n = 1 and x is positive (but less than 1), then x^n > x^(n+1). But if n = 1 and x is negative, then x^n > x^(n+1).

Statement 2: INSUFFICIENT. We know that n is even, so we know that x^n > 0, and therefore we can rephrase the question as “Is 1 > x?” However, we do not know the answer to that question.

Statements 1 & 2 TOGETHER: SUFFICIENT. Using Statement (2), we can rephrase the question as “Is 1 > x?”, to which Statement (1) gives us a definitive answer.

The answer is C: BOTH statements TOGETHER are sufficient to answer the question, but neither statement alone is sufficient.