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In Exercises 47 - 49, add or subtract terms whenever possible. 4√72 - 2√48
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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 50
Rationalize the denominator.
Guida verificata passo dopo passo1
Identify the expression to rationalize: \(\frac{3}{3+\sqrt{7}}\) where the denominator contains a sum with a square root.
Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(3 + \sqrt{7}\) is \(3 - \sqrt{7}\). So multiply by \(\frac{3 - \sqrt{7}}{3 - \sqrt{7}}\).
Apply the multiplication: The numerator becomes \(3 \times (3 - \sqrt{7})\) and the denominator becomes \((3 + \sqrt{7})(3 - \sqrt{7})\).
Use the difference of squares formula for the denominator: \((a + b)(a - b) = a^2 - b^2\). Here, \(a = 3\) and \(b = \sqrt{7}\), so the denominator simplifies to \(3^2 - (\sqrt{7})^2\).
Simplify the denominator by calculating \(3^2 = 9\) and \((\sqrt{7})^2 = 7\), so the denominator becomes \(9 - 7\). The numerator remains as \(3(3 - \sqrt{7})\).
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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. Typically, this is achieved by multiplying the numerator and denominator by a conjugate or an appropriate radical expression.
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Rationalizing Denominators
Conjugates of Binomials
The conjugate of a binomial expression like (a + √b) is (a - √b). Multiplying a binomial by its conjugate results in a difference of squares, which removes the square root terms. This property is essential for rationalizing denominators containing sums or differences involving square roots.
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Difference of Squares Formula
The difference of squares formula states that (x + y)(x - y) = x² - y². When applied to conjugates, it helps eliminate radicals by turning expressions like (3 + √7)(3 - √7) into 3² - (√7)² = 9 - 7 = 2, a rational number. This simplification is key to rationalizing denominators.
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04:14Special Products - Square Formulas
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