Problem 123 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Rationalizing the denominator means to get rid of any square roots in the denominator. For example, \(\frac{1}{\sqrt{5}}\) can be rationalized to \(\frac{\sqrt{5}}{5}\) and \(\frac{1}{5+\sqrt{5}}\) can be rationalized to \(\frac{5 - \sqrt{5}}{20}\).

Step 1: Rationalize the denominator of the first example

In the first example \(\frac{1}{\sqrt{5}}\), rationalizing the denominator involves multiplying both the numerator and the denominator by \(\sqrt{5}\). This gives: \(\frac{1* \sqrt{5}}{\sqrt{5}*\sqrt{5}}\). Simplifying this results in \(\frac{\sqrt{5}}{5}\).

Step 2: Rationalize the denominator of the second example

In the second example \(\frac{1}{5 + \sqrt{5}}\), to rationalize the denominator, use the method of multiplying by the conjugate. The conjugate of the denominator \(5 + \sqrt{5}\) is \(5 - \sqrt{5}\). Therefore, multiply both, the numerator and the denominator, by \(5 - \sqrt{5}\). This results in: \(\frac{1*(5 - \sqrt{5})}{(5 + \sqrt{5})*(5 - \sqrt{5})}\). This simplifies to \(\frac{5 - \sqrt{5}}{5^2 - (\sqrt{5})^2}\), further simplifying to \(\frac{5 - \sqrt{5}}{20}\).

Step 3: Final explanation

Rationalizing a denominator basically means transforming it into a rational number if it currently contains an irrational number. It is achieved by multiplying by an appropriate form of 1 (which is a number divided by itself), leading to the elimination of the square root in the denominator. It is a common operation in algebra, particularly when working with irrational roots.