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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2:19 minutes

Problem 76

Textbook Question

In Exercises 75–92, rationalize each denominator. Simplify, if possible.15----------√6 + 1

Verified step by step guidance

Identify the expression to rationalize: \( \frac{15}{\sqrt{6} + 1} \).

Multiply the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{6} - 1 \).

Write the expression as: \( \frac{15(\sqrt{6} - 1)}{(\sqrt{6} + 1)(\sqrt{6} - 1)} \).

Simplify the denominator using the difference of squares formula: \((a + b)(a - b) = a^2 - b^2\), where \(a = \sqrt{6}\) and \(b = 1\).

Simplify the numerator by distributing 15: \(15\sqrt{6} - 15\).

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Key Concepts

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

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any irrational numbers from the denominator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by a suitable expression that will result in a rational number in the denominator. For example, if the denominator is a binomial involving a square root, you can multiply by the conjugate of that binomial.

Conjugates

The conjugate of a binomial expression is formed by changing the sign between the two terms. For instance, the conjugate of (a + b) is (a - b). When multiplying a binomial by its conjugate, the result is a difference of squares, which eliminates the square root in the denominator, making it easier to simplify the expression.

Simplifying Radicals

Simplifying radicals involves reducing a square root to its simplest form. This can include factoring out perfect squares from under the radical sign and rewriting the expression. For example, √12 can be simplified to 2√3, as 4 is a perfect square factor of 12. This process is essential for presenting the final answer in a clear and concise manner.