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Chapter 10: Problem 126

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 refers to the process of removing any irrational numbers from the denominator of a fraction. \(\frac{1}{\sqrt{5}}\) rationalizes to \(\frac{\sqrt{5}}{5}\) and \(\frac{1}{5+\sqrt{5}}\) rationalizes to \(\frac{5 - \sqrt{5}}{20}\).

Step by step solution

01

Definition of Rationalizing a Denominator

Rationalizing the denominator of a fraction means removing any square roots or cube roots from the denominator. This is typically done by multiplying the numerator and the denominator of the fraction by the conjugate of the denominator.
02

Rationalize the Denominator of \(\frac{1}{\sqrt{5}}\)

Rationalizing the denominator of \(\frac{1}{\sqrt{5}}\) means removing \(\sqrt{5}\) from the denominator. This can be achieved by multiplying both the top and bottom of the fraction by \(\sqrt{5}\), which gives \(\frac{\sqrt{5}}{5}\).
03

Rationalize the Denominator of \(\frac{1}{5+\sqrt{5}}\)

To rationalize the denominator of \(\frac{1}{5+\sqrt{5}}\), we use the conjugate which is \(5 - \sqrt{5}\). By multiplying both the numerator and denominator by this conjugate, the resulting fraction is \(\frac{5 - \sqrt{5}}{20}\).

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Most popular questions from this chapter

In Exercises \(39-64,\) rationalize each denominator. $$\frac{5}{\sqrt[4]{x}}$$

Explain why \(\sqrt{x}=-1\) has no solution.

In Exercises \(129-132\), determine if each operation is performed correctly by graphing the function on each side of the equation with your graphing utility. Use the given viewing rectangle. The graphs should be the same. If they are not, correct the right side of the equation and then use your graphing utility to verify the correction. $$\begin{aligned} &\frac{3}{\sqrt{x+3}-\sqrt{x}}=\sqrt{x+3}+\sqrt{x}\\\ &[0,8,1] \text { by }[0,6,1] \end{aligned}$$

In Exercises \(105-112,\) add or subtract as indicated. Begin by rationalizing denominators for all terms in which denominators contain radicals. $$\sqrt{15}-\sqrt{\frac{5}{3}}+\sqrt{\frac{3}{5}}$$

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{\sqrt{11}-\sqrt{5}}{\sqrt{11}+\sqrt{5}}$$
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