Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Radical Expressions

College Algebra

0. Review of Algebra

Radical Expressions

Previous problemNext problem

1:08 minutes

Problem 10

Textbook Question

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

Step 1: Understand the notation of negative exponents. A negative exponent, such as \( a^{-n} \), means \( \frac{1}{a^n} \).

Step 2: Evaluate each expression in Column I using the rule for negative exponents.

Step 3: For expression (a) \( 5^{-3} \), rewrite it as \( \frac{1}{5^3} \).

Step 4: For expression (b) \( -5^{-3} \), rewrite it as \( -\frac{1}{5^3} \).

Step 5: For expression (c) \( (-5)^{-3} \), rewrite it as \( \frac{1}{(-5)^3} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the absolute value of the exponent. For example, a^-n = 1/a^n. This concept is crucial for simplifying expressions like 5^-3, which equals 1/(5^3) = 1/125.

Exponentiation of Negative Numbers

When raising a negative number to an exponent, the result depends on whether the exponent is even or odd. For instance, (-5)^3 results in -125, while (-5)^2 results in 25. Understanding this distinction is essential for evaluating expressions like (-5)^-3.

Simplifying Expressions with Parentheses

Parentheses in mathematical expressions dictate the order of operations. For example, -(-5)^-3 requires evaluating (-5)^-3 first, which is 1/(-5)^3 = -1/125, and then applying the negative sign outside the parentheses. This concept is vital for correctly interpreting and simplifying expressions.