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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1:31 minutes

Problem 28

Textbook Question

In Exercises 21–32, simplify by factoring.__³√54

Verified step by step guidance

Identify the expression to simplify: \( \sqrt[3]{54} \).

Factor 54 into its prime factors: \( 54 = 2 \times 3^3 \).

Apply the property of cube roots: \( \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b} \).

Separate the cube root into two parts: \( \sqrt[3]{2} \times \sqrt[3]{3^3} \).

Simplify \( \sqrt[3]{3^3} \) to 3, resulting in \( 3 \times \sqrt[3]{2} \).

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Key Concepts

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

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In algebra, this often involves identifying common factors, applying the distributive property, or recognizing special products such as squares or cubes. Understanding how to factor is essential for simplifying expressions and solving equations.

Cube Roots

A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3, since 3 × 3 × 3 = 27. In the context of simplifying expressions, recognizing cube roots can help in factoring expressions that involve cubic terms or in simplifying radical expressions.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, and can often be simplified by factoring. Simplifying radical expressions requires an understanding of how to manipulate the radicand (the number under the root) and apply properties of exponents. This is crucial for solving equations that include radicals and for simplifying complex expressions.