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Rewrite each expression without absolute value bars. |12 - π|
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Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 1, Problema 54
Rationalize the denominator.
Guida verificata passo dopo passo1
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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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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02:58
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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04:14Special Products - Square Formulas
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