Textbook Question
In Exercises 39–48, factor the difference of two squares. 16x4−81
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Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 1, Problem 47
Rationalize the denominator.
Verified step by step guidance1
Identify the expression to rationalize: \(\frac{\sqrt{2}}{\sqrt{5}}\). The goal is to eliminate the square root from the denominator.
Multiply both the numerator and the denominator by \(\sqrt{5}\), which is the conjugate in this case, to rationalize the denominator. This gives: \(\frac{\sqrt{2}}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}\).
Use the property of square roots that \(\sqrt{a} \times \sqrt{a} = a\) to simplify the denominator: \(\sqrt{5} \times \sqrt{5} = 5\).
Multiply the numerators together: \(\sqrt{2} \times \sqrt{5} = \sqrt{2 \times 5} = \sqrt{10}\).
Write the new expression with the rationalized denominator: \(\frac{\sqrt{10}}{5}\). This is the simplified form with no radical in the denominator.
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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 radicals (square roots) from the denominator of a fraction. This is done by multiplying the numerator and denominator by a suitable expression that will remove the radical from the denominator, making the expression easier to interpret and use.
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02:58
Rationalizing Denominators
Properties of Square Roots
Square roots have properties such as \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \) and \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). Understanding these properties helps simplify expressions involving radicals and is essential when rationalizing denominators.
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Multiplying by a Form of One
To rationalize a denominator, multiply the fraction by a form of one that contains the radical in the denominator, such as \( \frac{\sqrt{5}}{\sqrt{5}} \). This does not change the value of the expression but helps eliminate the radical from the denominator.
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Related Practice
Textbook Question
Add or subtract terms whenever possible. 7√5 + 13√5
Textbook Question
In Exercises 15–58, find each product. (x−3)2
Textbook Question
In Exercises 45–54, rationalize the denominator. 2/√10
Textbook Question
Simplify each exponential expression in Exercises 23–64.
Textbook Question
In Exercises 15–58, find each product. (4x2−1)2
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