Which statement about the base of (blah/(x+blah))^20 is true?

• A. The base is always positive.
• B. The base is always negative.
• C. The base is always nonzero.
• D. The base can be zero or nonzero depending on blah and x.

Answer: (D)

The base is the fraction blah divided by (x plus blah). Its value can vary depending on the actual values of blah and x; it is not fixed to be always positive or always negative.

If blah is zero (and the denominator isn’t zero), the base is zero, so the whole expression becomes zero after raising to the 20th power. If blah and x have the same sign relative to the denominator, the base can be positive; if they have opposite signs, the base can be negative. Since the denominator could be zero for some (blah, x) values, the base is undefined there as well, but within the domain where it’s defined, the base can be zero or a nonzero number.

Thus the only statement that correctly captures this variable possibility is that the base can be zero or nonzero depending on blah and x.