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Chapter 7: Problem 62

Rationalize each denominator. If possible, simplify your result. $$ \frac{5}{4-\sqrt{5}} $$

Short Answer

Expert verified
\( \frac{20 + 5\sqrt{5}}{11} \)

Step by step solution

01

Identify the Conjugate

To rationalize the denominator, identify the conjugate of the denominator. The conjugate of \(4 - \sqrt{5}\) is \(4 + \sqrt{5}\).
02

Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate of the denominator \[ \frac{5}{4 - \sqrt{5}} \times \frac{4 + \sqrt{5}}{4 + \sqrt{5}} \].
03

Expand the Numerator

Expand the numerator by distributing: \[ 5 \times (4 + \sqrt{5}) = 20 + 5\sqrt{5} \].
04

Expand the Denominator

Use the difference of squares formula \((a - b)(a + b) = a^2 - b^2\):\[ (4 - \sqrt{5})(4 + \sqrt{5}) = 4^2 - (\sqrt{5})^2 = 16 - 5 = 11 \].
05

Form the Rationalized Fraction

Combine the expanded numerator and denominator to form:\[ \frac{20 + 5\sqrt{5}}{11} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conjugate in Algebra
Rationalizing the denominator often involves using the conjugate of the denominator. This technique helps remove any irrational numbers. Conjugates are pairs of binomials that look alike except for the sign between the terms. For example, the conjugate of \(4 - \sqrt{5}\) is \(4 + \sqrt{5}\).
It's essential to multiply both the numerator and the denominator by this conjugate to maintain the fraction's value, as shown in the step-by-step solution.
Difference of Squares
The difference of squares formula is a valuable tool in algebra when rationalizing denominators. The formula states that \((a - b)(a + b) = a^2 - b^2\).
When you multiply two binomials, one being the conjugate of the other, you get the difference of their squares. In our example, we have \((4 - \sqrt{5})(4 + \sqrt{5})\), which simplifies to \4^2 - (\sqrt{5})^2 = 16 - 5 = 11\.
This step is critical as it removes the square root from the denominator.
Simplifying Fractions
After rationalizing the denominator, the next step is to simplify the fraction if possible. Simplifying a fraction involves reducing it to its lowest terms鈥攖he simplest form of the numerator and the denominator.
In the exercise, we have the expression \[\frac{20 + 5\sqrt{5}}{11}\]\

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