- When a negative base is raised to an integer power, the sign of the result depends on whether the exponent is even or odd.
- Multiplying a negative number an even number of times results in a positive value.
- Multiplying a negative number an odd number of times results in a negative value.
- The correct option is: \(\boxed{a}\).
Explanation
- Analyzing the Power Expression with a Negative Base
Consider a power expression with a negative base: \((-b)^n\), where \(b > 0\) so that \((-b)\) is a negative number, and \(n\) is an integer exponent. To determine the sign of the result, we analyze the effect of repeatedly multiplying the negative base by itself.
- Observing the Pattern of Signs
We examine the first few powers of a negative base \((-b)\):
- If \(n = 1\), \((-b)^1 = -b\), which is negative.
- If \(n = 2\), \((-b)^2 = (-b) \times (-b) = b^2\), which is positive.
- If \(n = 3\), \((-b)^3 = (-b)^2 \times (-b) = b^2 \times (-b) = -b^3\), which is negative.
- If \(n = 4\), \((-b)^4 = (-b)^3 \times (-b) = (-b^3) \times (-b) = b^4\), which is positive.
The pattern is clear: multiplying by the negative base \((-b)\) flips the sign of the product at each step. An even number of negative factors results in a positive product because each pair of negative factors cancels out \((- \times - = +)\). An odd number of negative factors leaves one negative sign remaining, making the product negative.
- Conclusion
Based on the pattern, we conclude the following rule:
- If the exponent \(n\) is even, the product of \(n\) negative numbers is positive.
- If the exponent \(n\) is odd, the product of \(n\) negative numbers is negative.
Comparing this to the provided options, option (a) states that an odd exponent results in a negative value and an even exponent results in a positive value. This aligns perfectly with our findings.
Examples
The sign rule for exponents is similar to checking whether you end up on the winning side of a game after an even or odd number of turns if each turn swaps your state. For example, if you start with a 'negative' debt of \$5 and multiply it by a factor of -1 each turn, after 2 turns you have a positive balance of \$5, but after 3 turns you are back to a negative debt of \$5. This rule helps in calculating compound interest or growth factors involving negative rates without needing to perform the full multiplication first.