Problem 112 - College Algebra

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

To rationalize a denominator, we want to eliminate any square roots or other irrational numbers. In the first example, we multiplied the numerator and the denominator by \(\sqrt{5}\), yielding \(\frac{\sqrt{5}}{5}\). In the second example, we multiplied by the conjugate, \((5 - \sqrt{5})\), resulting in \(\frac{5 - \sqrt{5}}{20}\). In both examples, the denominator is now a rational number.

Step 1: Understanding Rationalization

Rationalization is the process of eliminating any square roots or any other irrational numbers in the denominator of a fraction. The goal is to simplify the fraction to a more readable and easier form. This is possible by multiplying both the numerator and the denominator by a certain expression that will get rid of the square root or the irrational number in the denominator.

Step 2: Rationalizing the Denominator - Example 1

Let's consider the example with the fraction \(\frac{1}{\sqrt{5}}\). To rationalize this, we multiply both the numerator and the denominator by \(\sqrt{5}\), the result is: \(\frac{1 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}}\), which simplifies to: \(\frac{\sqrt{5}}{5}\). Our denominator is now a rational number.

Step 3: Rationalizing the Denominator - Example 2

Now, let's take the more complex example, \(\frac{1}{5+\sqrt{5}}\). To simplify this, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \((5 + \sqrt{5})\) is \((5 - \sqrt{5})\). The result is: \(\frac{1 \times (5 - \sqrt{5})}{(5 + \sqrt{5}) \times (5 - \sqrt{5})}\). By applying the difference of squares formula, the denominator simplifies to \((5^2 - (\sqrt{5})^2)\), or \((25 - 5)\), which further simplifies to \(20\). Hence, our fraction simplifies to: \(\frac{5 - \sqrt{5}}{20}\). Our denominator is again a rational number.