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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1:39 minutes

Problem 65

Textbook Question

Evaluate each expression 125^(-1/3)

Verified step by step guidance

Recognize that the expression \( 125^{-1/3} \) involves a negative exponent and a fractional exponent.

Recall that a negative exponent indicates the reciprocal of the base raised to the positive of that exponent. So, \( 125^{-1/3} = \frac{1}{125^{1/3}} \).

Understand that the fractional exponent \( 1/3 \) represents the cube root. Therefore, \( 125^{1/3} \) is the cube root of 125.

Identify that 125 is a perfect cube, specifically \( 5^3 = 125 \).

Conclude that \( 125^{1/3} = 5 \), so \( 125^{-1/3} = \frac{1}{5} \).

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Key Concepts

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

Exponents and Radicals

Exponents represent repeated multiplication of a base number. A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For example, a base of 125 with an exponent of -1/3 means we take the reciprocal of 125 raised to the power of 1/3, which involves both negative and fractional exponent rules.

Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In this case, 125 is a perfect cube, as 5 x 5 x 5 equals 125. Therefore, the cube root of 125 is 5, which is essential for evaluating the expression 125^(-1/3).

Reciprocal

The reciprocal of a number is 1 divided by that number. When evaluating expressions with negative exponents, finding the reciprocal is crucial. For instance, the expression 125^(-1/3) translates to 1 divided by the cube root of 125, which simplifies the evaluation process and leads to the final result.