Match each expression in Column I with its equivalent expression in Column II. Choices may be used once, more than once, or not at all. a. 5^-3 b. -5^-3 c. (-5)^-3 d. -(-5)^-3 A. 125 B. -125 C. 1/125 D. -1/125

Verified step by step guidance

1. First, recall the meaning of negative exponents: for any nonzero number \(a\) and positive integer \(n\), \(a^{-n} = \frac{1}{a^n}\).
2. Evaluate each expression in Column I by applying the exponent rules carefully, paying attention to the placement of parentheses and the negative signs.
3. For expression (a) \$5^{-3}\(, rewrite it as \)\frac{1}{5^3}\( and recognize that \)5^3 = 125\(, so this expression equals \)\frac{1}{125}$.
4. For expression (b) \(-5^{-3}\), note that the negative sign is outside the exponent, so it is \(- (5^{-3}) = - \frac{1}{125}\).
5. For expression (c) \((-5)^{-3}\), apply the negative exponent to the entire base \(-5\), so it becomes \(\frac{1}{(-5)^3}\). Since \((-5)^3 = -125\), this equals \(\frac{1}{-125} = -\frac{1}{125}\).

Key Concepts

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, a^-n = 1/a^n. This concept helps in rewriting expressions like 5^-3 as 1/5^3.

Order of Operations and Negative Signs

Understanding how negative signs interact with exponents is crucial. For instance, -5^-3 means the negative of 5^-3, while (-5)^-3 means the entire base -5 is raised to the negative exponent. Parentheses affect the base and thus the result.

Evaluating Powers of Negative Numbers

Raising a negative number to an odd power results in a negative number, while raising it to an even power results in a positive number. This affects the sign of expressions like (-5)^-3, which equals 1/(-5)^3 = -1/125.