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

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1: minutes

Problem 13

Textbook Question

Write each root using exponents and evaluate. ∜81

Verified step by step guidance

Recognize that the expression ∜81 represents the fourth root of 81.

Rewrite the fourth root using exponents: 81^{1/4}.

Express 81 as a power of a smaller number: 81 = 3^4.

Substitute 3^4 for 81 in the expression: (3^4)^{1/4}.

Apply the power of a power property: (3^4)^{1/4} = 3^{(4 * 1/4)} = 3^1.

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Key Concepts

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

Roots and Exponents

Roots and exponents are fundamental concepts in algebra that describe the relationship between numbers. The nth root of a number is a value that, when raised to the nth power, gives the original number. For example, the square root of 9 is 3 because 3^2 = 9. This relationship can also be expressed using fractional exponents, where the nth root of a number 'a' is represented as a^(1/n).

Evaluating Roots

Evaluating roots involves finding the value that satisfies the root equation. For instance, to evaluate the fourth root of 81, we seek a number that, when raised to the fourth power, equals 81. This process may involve recognizing perfect powers or using prime factorization to simplify the evaluation. In this case, since 81 = 3^4, the fourth root is 3.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves reducing complex expressions to their simplest form. This can include factoring, combining like terms, and applying properties of exponents and roots. For example, when dealing with roots, one can express them in terms of exponents to facilitate easier calculations and comparisons, such as rewriting ∜81 as 81^(1/4).