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Chapter 8: Problem 86

Describe what it means to rationalize a denominator. Use both \(\frac{1}{\sqrt{5}}\) and \(\frac{1}{5+\sqrt{5}}\) in your explanation.

Short Answer

Expert verified
The term 'rationalizing the denominator' refers to the process of eliminating radicals from the denominator of a fraction, achieved by multiplying the numerator and denominator by an appropriate term. By this process, \(\frac{1}{\sqrt{5}}\) becomes \(\frac{\sqrt{5}}{5}\) and \(\frac{1}{5+\sqrt{5}}\) becomes \(\frac{5 - \sqrt{5}}{20}\).

Step by step solution

01

Understanding Rationalizing the Denominator

Rationalizing the denominator refers to the process of eliminating any radical expressions (such as roots) from the denominator of a fraction. It is carried out by multiplying the numerator and the denominator by a conjugate or the radical expression itself.
02

Rationalize \(\frac{1}{\sqrt{5}}\)

To begin, take the fraction \(\frac{1}{\sqrt{5}}\). You rationalize this fraction by multiplying both the numerator and the denominator by \(\sqrt{5}\). This gives the fraction \(\frac{\sqrt{5}}{5}\). This fraction is the rationalized version of \(\frac{1}{\sqrt{5}}\), because the denominator is now a rational number.
03

Rationalize \(\frac{1}{5+\sqrt{5}}\)

Next, consider the fraction \(\frac{1}{5+\sqrt{5}}\). To rationalize this fraction, multiply both the numerator and the denominator by the conjugate of the denominator, which is \(5 - \sqrt{5}\). This results in the fraction \(\frac{1 * (5 - \sqrt{5})}{(5+\sqrt{5}) * (5 - \sqrt{5})}\), which simplifies to \(\frac{5 - \sqrt{5}}{5^2 - (\sqrt{5})^2}\) or \(\frac{5 - \sqrt{5}}{20}\). This is the rationalized version of \(\frac{1}{5+\sqrt{5}}\), as the denominator is now a rational number.

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