Problem 123 - College Algebra Essentials

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 means to eliminate any irrational numbers present in it through multiplication by a suitable expression. The rationalized denominators for \(\frac{1}{\sqrt{5}}\) and \(\frac{1}{5+\sqrt{5}}\) are \(\frac{\sqrt{5}}{5}\) and \(\frac{5 - \sqrt{5}}{20}\) respectively.

Step 1: Rationalize the denominator of \(\frac{1}{\sqrt{5}}\)

To rationalize the denominator of \(\frac{1}{\sqrt{5}}\), multiply the numerator and denominator by \(\sqrt{5}\). This gives \(\frac{\sqrt{5}}{5}\).

Step 2: Rationalize the denominator of \(\frac{1}{5+\sqrt{5}}\)

For the expression \(\frac{1}{5+\sqrt{5}}\), the conjugate of the denominator (5 + \(\sqrt{5}\)) is needed. This is obtained by changing the sign in the middle, producing (5 - \(\sqrt{5}\)). Multiply the numerator and the denominator by this conjugate to get \(\frac{5 - \sqrt{5}}{20}\). Rationalizing this expression eliminates the square root from the denominator.

Step 3: Simplify the expressions

Both expressions \(\frac{\sqrt{5}}{5}\) and \(\frac{5 - \sqrt{5}}{20}\) are now rationalized. They may be left as they are or simplified further if possible. In this case, the expressions are already at their simplest form.