Rationalize the denominator. (a) (6)/(√(7))(b) √((8)/(45))(c) (12)/(√(3 p))

Name: Problem 261, Chapter 8: Intermediate Algebra
Uploaded: 2022-09-27T19:47:18-08:00
Duration: 1 min 51 s
Channel: Natalie Belcher
Description: Rationalize the denominator. (a) (6)/(√(7))(b) √((8)/(45))(c) (12)/(√(3 p))

Step 1:u000ATo rationalize the denominator, we multiply both the numerator and the denominator by th

Rationalize the denominator. (a) (6)/(√(7))(b)...

Key Concepts

Rationalizing the Denominator

This concept refers to the process of eliminating irrational numbers, such as square roots, from the denominator of a fraction. The idea is to rewrite the expression in an equivalent form where the denominator is a rational number. This is often accomplished by multiplying both the numerator and denominator by an appropriate radical expression, effectively using the property that multiplying by 1 (in a disguised form) keeps the value unchanged.

Square Root Properties

Understanding the properties of square roots is essential when rationalizing denominators. Key properties include the multiplication and division rules, where ?a * ?b = ?(ab) and ?(a/b) = ?a/?b. These rules allow you to combine and simplify radicals, facilitating the process of eliminating radicals from the denominators.

Multiplicative Identity and Form of One

The concept of the multiplicative identity is used in rationalization by multiplying by a form of one that contains a radical. For example, if you have a denominator of ?b, multiplying numerator and denominator by ?b yields (?b/?b) which equals one. This preserves the value of the expression while transforming the denominator into a rational number.

Simplifying Radical Expressions

After rationalizing the denominator, it is often necessary to simplify the resulting radical expressions by identifying and extracting perfect squares. This step ensures that the expression is in its simplest form, making it easier to interpret and work with further. Recognizing factors that are perfect squares in the radicand allows for simplification and cleaner final answers.

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