Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problema 75a

Capitolo 2, Problema 75a

Exercises 73–75 will help you prepare for the material covered in the next section. Rationalize the denominator: (7 + 4√2)/(2 - 5√2).

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Identify the problem: The goal is to rationalize the denominator of the given expression $(7 + 4\sqrt{2}) / (2 - 5\sqrt{2})$. This means eliminating the square root from the denominator by multiplying both numerator and denominator by the conjugate of the denominator.

Determine the conjugate of the denominator: The conjugate of $2 - 5\sqrt{2}$ is $2 + 5\sqrt{2}$. Multiplying by the conjugate will use the difference of squares formula $(a - b)(a + b) = a^2 - b^2$ to eliminate the square root.

Multiply both numerator and denominator by the conjugate: Multiply $(7 + 4\sqrt{2})$ by $(2 + 5\sqrt{2})$ in the numerator, and $(2 - 5\sqrt{2})$ by $(2 + 5\sqrt{2})$ in the denominator. This ensures the value of the expression remains unchanged.

Simplify the denominator: Use the difference of squares formula $(a - b)(a + b) = a^2 - b^2$ to simplify $(2 - 5\sqrt{2})(2 + 5\sqrt{2})$. This will result in $2^2 - (5\sqrt{2})^2$.

Simplify the numerator: Expand $(7 + 4\sqrt{2})(2 + 5\sqrt{2})$ using the distributive property (FOIL method). Combine like terms, ensuring to simplify any terms involving $\sqrt{2}$. After simplifying both numerator and denominator, the expression will be rationalized.

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Rationalizing the Denominator

Rationalizing the denominator involves eliminating any irrational numbers from the denominator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by a suitable expression that will result in a rational number in the denominator. For example, if the denominator contains a square root, multiplying by the conjugate can help achieve this.

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Rationalizing Denominators

Conjugates

The conjugate of a binomial expression is formed by changing the sign of the second term. For instance, the conjugate of (a + b) is (a - b). When multiplying a binomial by its conjugate, the result is a difference of squares, which eliminates the square root in the denominator, making it easier to rationalize.

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Complex Conjugates

Simplifying Radicals

Simplifying radicals involves reducing a square root or other root to its simplest form. This can include factoring out perfect squares from under the radical sign or rewriting the expression in a way that minimizes the radical's complexity. Understanding how to simplify radicals is essential for effectively rationalizing denominators and simplifying expressions.

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Adding & Subtracting Unlike Radicals by Simplifying

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