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

Exponents

College Algebra

0. Review of Algebra

Exponents

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2:07 minutes

Problem 26

Textbook Question

Evaluate each exponential expression: 5^(-3) • 5

Verified step by step guidance

Identify the base and the exponents in the expression: The base is 5, and the exponents are -3 and 1 (since 5 is the same as 5^1).

Apply the property of exponents that states when multiplying like bases, you add the exponents: \(5^{-3} \cdot 5^1 = 5^{-3+1}\).

Simplify the exponent: \(-3 + 1 = -2\), so the expression becomes \(5^{-2}\).

Recall the rule for negative exponents: \(a^{-n} = \frac{1}{a^n}\).

Apply the rule for negative exponents to rewrite \(5^{-2}\) as \(\frac{1}{5^2}\).

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Key Concepts

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

Exponential Notation

Exponential notation is a mathematical shorthand used to represent repeated multiplication of a number by itself. In the expression a^n, 'a' is the base and 'n' is the exponent, indicating how many times 'a' is multiplied. For example, 5^3 means 5 multiplied by itself three times, resulting in 125.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the absolute value of the exponent. For instance, 5^(-3) is equivalent to 1/(5^3), which simplifies to 1/125. This concept is crucial for simplifying expressions involving negative exponents and understanding their impact on calculations.

Product of Powers Property

The Product of Powers Property states that when multiplying two expressions with the same base, you can add their exponents. For example, a^m • a^n = a^(m+n). In the given expression, 5^(-3) • 5 can be rewritten as 5^(-3) • 5^1, allowing us to combine the exponents to simplify the expression to 5^(-2).