Explanation

This question can be computed directly, knowing that negative numbers turn positive in pairs, and so will be positive when taken to even exponents and negative when taken to odd powers. Also, when dealing with negative signs, we can proceed with organization and confidence by putting everything first in parentheses.







The correct answer is (E).

By the way, if the zeroth power puzzles you, consider this. In the fraction , the four 3's that are common to the numerator and denominator cancel, leaving the difference, . Similarly, in the case of , the same thing happens, but the extra three is left in the denominator: so . This provides a concrete example of why it makes sense that negative exponents correspond to positive exponents in the denominator. Secondly, it provides a concrete example of why we subtract the exponents when we have exponents of the same base in the numerator and the denominator. Finally, in the case of , the number of 3's is five up top and on the bottom, so everything will cancel, yielding . Since the base 3 is not material to these outcomes and could be replaced by any number or a variable, we have made a mini-proof of why these exponent properties are true and make sense.

Again, the correct answer is (E).