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.

Verified step by step guidance

1. Step 1: Understand the meaning of negative exponents. Recall that for any nonzero number \(a\) and positive integer \(n\), \(a^{-n} = \frac{1}{a^n}\). This means a negative exponent indicates the reciprocal of the base raised to the positive exponent.
2. Step 2: Evaluate each expression in Column I by applying the negative exponent rule and considering the sign carefully. For example, for \$5^{-3}\(, rewrite it as \)\frac{1}{5^3}$.
3. Step 3: Calculate the value of \$5^3\( which is \)5 \times 5 \times 5\(. This will help you express \)5^{-3}$ as a fraction.
4. Step 4: For expressions with a negative sign outside the base, such as \(-5^{-3}\), remember the negative sign applies after evaluating \$5^{-3}\(. For expressions like \)(-5)^{-3}\(, the negative sign is part of the base, so raise \)-5\( to the power of \)-3$.
5. Step 5: Match each simplified expression from Column I to the equivalent value in Column II by comparing their numerical values and signs.

Key Concepts

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, a⁻³ equals 1 divided by a³. This rule helps simplify expressions like 5⁻³ to 1/125.

Order of Operations and Sign Placement

Understanding how negative signs interact with exponents is crucial. For instance, -5⁻³ means the negative of 5⁻³, while (-5)⁻³ means the reciprocal of -5 cubed. Parentheses determine whether the negative sign is part of the base or applied afterward.

Evaluating Powers of Negative Numbers

Raising a negative number to an odd power results in a negative number, while an even power yields a positive number. For example, (-5)³ = -125, so (-5)⁻³ = 1/(-125) = -1/125. This concept helps match expressions with their correct values.