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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1:53 minutes

Problem 102

Textbook Question

In Exercises 101–108, simplify by reducing the index of the radical. ⁴√7^2

Verified step by step guidance

Recognize that the expression \( \sqrt[4]{7^2} \) involves a radical with an index of 4.

Recall the property of radicals: \( \sqrt[n]{a^m} = a^{m/n} \).

Apply this property to the expression: \( \sqrt[4]{7^2} = 7^{2/4} \).

Simplify the exponent \( 2/4 \) by reducing the fraction to its simplest form.

Rewrite the expression using the simplified exponent.

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Key Concepts

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

Radical Expressions

Radical expressions involve roots, such as square roots or fourth roots. The expression ⁴√7² represents the fourth root of 7 squared. Understanding how to manipulate these expressions is crucial for simplification, as it allows us to rewrite them in a more manageable form.

Index of a Radical

The index of a radical indicates the degree of the root being taken. In the expression ⁴√7², the index is 4, meaning we are looking for a number that, when raised to the fourth power, equals 7². Reducing the index involves simplifying the expression to make calculations easier.

Properties of Exponents

Properties of exponents govern how to simplify expressions involving powers. For instance, the property a^(m/n) = n√(a^m) allows us to express roots in terms of exponents. This is essential for simplifying radical expressions, as it helps in rewriting them in a form that is easier to evaluate or manipulate.