Final answer:
The counterexample to the statement that cubing any negative integer results in a value greater than zero is -1³ = -1. This demonstrates that a negative number cubed will indeed result in a negative number, disproving the original claim.
Step-by-step explanation:
The statement in question posits that when a negative integer is cubed, the result is always greater than zero. To find a counterexample, one must look for a case where cubing a negative integer results in a value that is not greater than zero. Let's examine the given options:
- 2³ = 8: This is cubing a positive integer, so it's not a counterexample.
- 5³ = 125: Again, this is a positive integer being cubed.
- -1³ = -1: This is the act of cubing a negative integer. Since the result is -1, which is not greater than zero, this serves as a counterexample.
- 1⁻³ = 1: This involves a negative exponent, not a negative base, and is not relevant to the counterexample search.
Therefore, the correct counterexample is -1³ = -1, which demonstrates that the original statement is false. Recall the multiplication rules for signs: when you multiply three of the same negative numbers, the result is negative because an odd number of negative factors produces a negative product. Consequently, any negative number to the power of three will also result in a negative number.
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