Question: All Sorts of Rules

Let's start from here:
QUANTITY A: x^(-1)
QUANTITY B: x^(-4)

At this point, I decided to apply the rule concerning numbers raised to ODD and EVEN powers. However, let's do what you did, and rewrite the powers as fractions.

When we do this we get:
QUANTITY A: 1/(x^1)
QUANTITY B: 1/(x^4)

Since we're told x is negative, let's rewrite as:
QUANTITY A: 1/(NEGATIVE^1)
QUANTITY B: 1/(NEGATIVE^4)

NEGATIVE^1 = NEGATIVE, and NEGATIVE^4 = POSITIVE. So, let's rewrite as:
QUANTITY A: 1/NEGATIVE
QUANTITY B: 1/POSITIVE

Simplify:
QUANTITY A: NEGATIVE
QUANTITY B: POSITIVE

Since a positive number is always greater than a negative number, Quantity B is greater.

Quote: "I got -1/2 for QA and 1/16 for QB, and concluded that that smaller negative fraction was the larger number"

That strategy is perfectly fine when BOTH fractions are NEGATIVE. For example, we know that -1/20 is greater than -1/7. However, in this question, one fraction is positive and one fraction in negative, in which case the POSITIVE fraction will ALWAYS be greater than the NEGATIVE fraction.

Does that help?

Cheers,
Brent

In your last step (when you multiply both quantities by x), you are breaking the rule that says NOT to multiply both quantities by a negative value (we're told that x < -1)

The problem is apparent when you get to:
QUANTITY A: 1
QUANTITY B: 1/(x^3)

Since x is negative, we know that x^3 is negative, so we get:
QUANTITY A: 1
QUANTITY B: 1/negative

Add this evaluates to:
QUANTITY A: 1
QUANTITY B: some negative value

In which case, quantity A is greater (which is incorrect).

Does that help?

Cheers,
Brent