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Ch. R - Algebra Review
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. R - Algebra ReviewProblem 139
Chapter 1, Problem 139

Simplify. See Example 9. (1/2)/(1 - (√5/2))

Verified step by step guidance
1
Rewrite the given expression clearly as a fraction: \(\frac{\frac{1}{2}}{1 - \frac{\sqrt{5}}{2}}\).
To simplify the complex fraction, multiply the numerator and denominator by 2 to eliminate the fractions inside the denominator: \(\frac{1}{2} \div \left(1 - \frac{\sqrt{5}}{2}\right) = \frac{1}{2} \times \frac{2}{2 - \sqrt{5}}\).
Simplify the multiplication in the numerator: \(\frac{1}{2} \times 2 = 1\), so the expression becomes \(\frac{1}{2 - \sqrt{5}}\).
Rationalize the denominator by multiplying numerator and denominator by the conjugate of the denominator, which is \(2 + \sqrt{5}\): \(\frac{1}{2 - \sqrt{5}} \times \frac{2 + \sqrt{5}}{2 + \sqrt{5}}\).
Use the difference of squares formula to simplify the denominator: \((2)^2 - (\sqrt{5})^2 = 4 - 5 = -1\). Then write the simplified expression as \(\frac{2 + \sqrt{5}}{-1}\) and simplify the negative sign.

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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 irrational numbers, such as square roots, from the denominator of a fraction. This is typically done by multiplying the numerator and denominator by a conjugate or an appropriate radical to create a rational denominator, simplifying the expression.
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Conjugates of Binomials

The conjugate of a binomial expression a + b is a - b, and vice versa. Multiplying a binomial by its conjugate results in the difference of squares, which removes the square root terms and simplifies the expression, a key step in rationalizing denominators.
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Simplifying Radical Expressions

Simplifying radical expressions involves reducing square roots to their simplest form by factoring out perfect squares. This process helps in making expressions easier to work with and is essential when performing operations like addition, subtraction, or rationalization.
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Related Practice
Textbook Question

For Individual or Group Work (Exercises 147 – 150)In calculus, it is sometimes desirable to rationalize a numerator. To do this, we multiply the numerator and the denominator by the conjugate of the numerator. For example, (6 - √2)/4 = (6 - √2)/4 × (6 + √2)/(6 + √2) = (36 - 2)/(4(6 + √2)) = 34/(4(6 + √2)) = 17/(2(6 + √2)) = 17/(6 + √2).

2√10 + √7 30

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Textbook Question

Simplify. See Example 9. (-√2/3)/(√7/3)

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Textbook Question

Simplify. See Example 9. (√7/5)/(√3/10)

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Textbook Question

For Individual or Group Work (Exercises 147 – 150)In calculus, it is sometimes desirable to rationalize a numerator. To do this, we multiply the numerator and the denominator by the conjugate of the numerator. For example, (6 - √2)/4 = (6 - √2)/4 × (6 + √2)/(6 + √2) = (36 - 2)/(4(6 + √2)) = 34/(4(6 + √2)) = 17/(2(6 + √2)) = 17/(6 + √2). Rationalize each numerator. (6 - √3)/8

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Textbook Question

Rationalize each denominator. See Example 8. (√2 - √3)/(√6 - √5)

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Textbook Question

Simplify. See Example 9. (√3/2)/(1 - (√3/2))

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