Textbook Question
Rewrite each expression without absolute value bars. |12 - π|
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 54
Rationalize the denominator.
Verified step by step guidance1
Identify the expression to rationalize: \(\frac{11}{\sqrt{7} - \sqrt{3}}\).
Recognize that the denominator is a binomial involving square roots, so multiply numerator and denominator by the conjugate of the denominator to rationalize it. The conjugate of \(\sqrt{7} - \sqrt{3}\) is \(\sqrt{7} + \sqrt{3}\).
Multiply numerator and denominator by the conjugate: \(\frac{11}{\sqrt{7} - \sqrt{3}} \times \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} + \sqrt{3}}\).
Apply the difference of squares formula to the denominator: \((a - b)(a + b) = a^2 - b^2\). Here, \(a = \sqrt{7}\) and \(b = \sqrt{3}\), so the denominator becomes \(7 - 3\).
Write the new expression as \(\frac{11(\sqrt{7} + \sqrt{3})}{7 - 3}\) and simplify the denominator to complete the rationalization process.
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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 to simplify the expression and make it easier to work with. For denominators with square roots, multiplying numerator and denominator by a suitable expression removes the radical.
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Rationalizing Denominators
Conjugates of Binomials
The conjugate of a binomial expression like (√7 − √3) is (√7 + √3). Multiplying a binomial by its conjugate uses the difference of squares formula, which eliminates the square roots in the denominator by producing a rational number.
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Difference of Squares Formula
The difference of squares formula states that (a − b)(a + b) = a² − b². This property is used to simplify expressions involving conjugates, especially when rationalizing denominators containing square roots, by turning the product into a difference of perfect squares.
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Related Practice
Textbook Question
Rewrite each expression without absolute value bars. |7 - π|
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Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number. ³√8
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In Exercises 50 - 53, rationalize the denominator. 14/(√7 - √5)
Textbook Question
Rationalize the denominator.
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Simplify each exponential expression in Exercises 23–64. (4x^3)^−2