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Chapter 0: Problem 123

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
Rationalizing the denominator of a fraction involves changing the form of the fraction so the square root is not in the denominator. For \(\frac{1}{\sqrt{5}}\), multiply numerator and denominator by \(\sqrt{5}\), which results in \(\frac{\sqrt{5}}{5}\). For \(\frac{1}{5+\sqrt{5}}\), multiply numerator and denominator by \(5 - \sqrt{5}\), which results in \(\frac{5 - \sqrt{5}}{20}\).

Step by step solution

01

Rationalizing a denominator with just a square root.

For the first fraction \(\frac{1}{\sqrt{5}}\), to rationalize the denominator we multiply both the numerator and the denominator by the radical in the denominator. So, \(\frac{1}{\sqrt{5}}\) * \(\frac{\sqrt{5}}{\sqrt{5}}\) = \(\frac{\sqrt{5}}{5}\). This removes the square root from the denominator.
02

Rationalizing a denominator with a binomial including a square root.

For the second fraction \(\frac{1}{5+\sqrt{5}}\), we need to use a different approach. Here, the denominator is a sum that involves a square root. To rationalize this denominator, multiply both the numerator and the denominator by the conjugate of the denominator. In this case, the conjugate is \(5 - \sqrt{5}\). So, \(\frac{1}{5+\sqrt{5}}\) * \(\frac{5-\sqrt{5}}{5-\sqrt{5}}\) = \(\frac{5 - \sqrt{5}}{25 - 5}\) which simplifies to \(\frac{5 - \sqrt{5}}{20}\). This process eliminated the square root from the denominator.
03

Summary of the process.

The process of removing the square root from the denominator, or rationalizing the denominator, involves multiplying the fraction by an appropriate form of 1; either by the square root of the denominator or by the conjugate of the denominator, depending on what the denominator contains. Doing this changes the denominator to a rational number while keeping the value of the fraction the same (since multiplying by any form of 1 does not change the value of an expression).

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Most popular questions from this chapter

Exercises \(142-144\) will help you prepare for the material covered in the next section. Simplify and express the answer in descending powers of \(x\) : $$ 2 x\left(x^{2}+4 x+5\right)+3\left(x^{2}+4 x+5\right) $$

Perform the indicated operations. $$(x-y)^{-1}+(x-y)^{-2}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I prefer interval notation over set-builder notation because it takes less space to write solution sets.

$$\begin{array}{l} \text { Find the exact value of } \sqrt{13+\sqrt{2}+\frac{7}{3+\sqrt{2}}} \text { without } \\ \text { the use of a calculator. } \end{array}$$

Perform the indicated operations. Simplify the result, if possible. $$\left(\frac{2 x+3}{x+1} \cdot \frac{x^{2}+4 x-5}{2 x^{2}+x-3}\right)-\frac{2}{x+2}$$
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