Final answer:
A positive integer raised to a negative odd power results in a positive fractional value because the negative exponent equates to the reciprocal of the number raised to the absolute value of the exponent.
Step-by-step explanation:
When a positive integer is raised to a negative odd power, it always results in a positive fractional value. This occurs because a negative exponent means that the number will be the reciprocal of the base raised to the absolute value of the exponent—essentially flipping the base to the denominator.
For example, if you have the expression 2-3, it would equal 1 / 23 or 1/8, which is a positive fraction. We know that two positive numbers multiply to a positive result, and similarly, when a positive number is raised to an even power, it remains positive. Since the negative exponent is turning it into a reciprocal, the resulting fraction is still positive.
The multiplication rules also remind us that a number with an even power, when multiplied by itself (an odd number of times), will always result in a positive product. For instance, (-3)2 x (-3) = 9 x (-3) = -27, which demonstrates the rule for multiplying numbers of opposite signs.