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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 141
Chapter 1, Problem 141

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

Verified step by step guidance
1
Identify the expression to simplify: \(\frac{\frac{\sqrt{3}}{2}}{1 - \frac{\sqrt{3}}{2}}\).
Rewrite the complex fraction as a division of two fractions: \(\frac{\sqrt{3}}{2} \div \left(1 - \frac{\sqrt{3}}{2}\right)\).
Simplify the denominator by expressing 1 as \(\frac{2}{2}\) to have a common denominator: \(1 - \frac{\sqrt{3}}{2} = \frac{2}{2} - \frac{\sqrt{3}}{2} = \frac{2 - \sqrt{3}}{2}\).
Rewrite the division as multiplication by the reciprocal: \(\frac{\sqrt{3}}{2} \times \frac{2}{2 - \sqrt{3}}\).
Cancel common factors if possible and then rationalize the denominator by multiplying numerator and denominator by the conjugate of the denominator, which is \(2 + \sqrt{3}\).

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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 simplify the expression.
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Conjugates in Algebra

The conjugate of a binomial expression a + b is a - b, and vice versa. Multiplying by the conjugate helps remove square roots from denominators because it uses the difference of squares formula, resulting in a rational number.
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Simplifying Square Roots and Fractions

Simplifying square roots involves expressing them in simplest radical form, while simplifying fractions means reducing them to their lowest terms. Combining these skills helps in rewriting expressions in a clearer, more manageable form.
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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

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

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