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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1:52 minutes

Problem 28

Textbook Question

In Exercises 1–38, multiply as indicated. If possible, simplify any radical expressions that appear in the product.(√6 + √2) (√6 - √2)

Verified step by step guidance

Recognize that the expression \((\sqrt{6} + \sqrt{2})(\sqrt{6} - \sqrt{2})\) is a difference of squares.

Apply the difference of squares formula: \((a + b)(a - b) = a^2 - b^2\).

Identify \(a = \sqrt{6}\) and \(b = \sqrt{2}\).

Calculate \(a^2 = (\sqrt{6})^2 = 6\) and \(b^2 = (\sqrt{2})^2 = 2\).

Subtract the squares: \(a^2 - b^2 = 6 - 2\).

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Key Concepts

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

Multiplying Binomials

Multiplying binomials involves applying the distributive property, often referred to as the FOIL method (First, Outside, Inside, Last). This technique helps in systematically multiplying each term in the first binomial by each term in the second binomial, ensuring that all combinations are accounted for. In the given expression, (√6 + √2)(√6 - √2), this method will yield a difference of squares.

Difference of Squares

The difference of squares is a specific algebraic identity that states a² - b² = (a + b)(a - b). This identity is particularly useful for simplifying expressions where one term is subtracted from another, both of which are perfect squares. In the context of the problem, recognizing that √6 and √2 are the 'a' and 'b' allows for a straightforward simplification of the product.

Simplifying Radical Expressions

Simplifying radical expressions involves reducing the expression to its simplest form, which often includes combining like terms and removing any perfect squares from under the radical sign. This process is essential for clarity and ease of further calculations. In the given problem, after applying the difference of squares, any resulting radical expressions should be simplified to ensure the final answer is presented in its simplest form.