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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
The rationalization of \(\frac{1}{\sqrt{5}}\) is \(\frac{\sqrt{5}}{5}\) and the rationalization of \(\frac{1}{5 + \sqrt{5}}\) is \(\frac{5 - \sqrt{5}}{20}\). Rationalization is the process used to make the denominator of a fraction become a rational number by multiplying both the numerator and denominator by the appropriate value which usually is the square root in the denominator or the conjugate of the denominator if it's a binomial.

Step by step solution

01

Understanding rationalization

Rationalization is the process of eliminating the root in the denominator of a fraction. It involves manipulating the fraction in such a way that the denominator becomes a rational number without changing the value of the fraction.
02

Rationalize \(\frac{1}{\sqrt{5}}\)

The fraction \(\frac{1}{\sqrt{5}}\) has an irrational denominator. To rationalize this, multiply both numerator and denominator by \(\sqrt{5}\), since \(\sqrt{5} \times \sqrt{5} = 5\) which is a rational number. So, we get: \(\frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5}\)
03

Rationalize \(\frac{1}{5 + \sqrt{5}}\)

The fraction \(\frac{1}{5 + \sqrt{5}}\) has an irrational denominator. We can rationalize it by multiplying both the numerator and the denominator by the conjugate of the denominator, which is \(5 - \sqrt{5}\), effectively applying the difference of squares formula. \(\frac{1}{5 + \sqrt{5}} \times \frac{5 - \sqrt{5}}{5 - \sqrt{5}} = \frac{5 - \sqrt{5}}{25 - 5}\)

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Most popular questions from this chapter

Simplify using properties of exponents. $$\left(7 x^{\frac{1}{3}}\right)\left(2 x^{\frac{1}{4}}\right)$$

Evaluate each expression. $$\sqrt[3]{\sqrt{\sqrt{169}+\sqrt{9}}+\sqrt{\sqrt[3]{1000}+\sqrt[3]{216}}}$$

If \(b^{A}=M N, b^{C}=M,\) and \(b^{D}=N,\) what is the relationship among \(A, C,\) and \(D ?\)

Simplify using properties of exponents. $$\frac{72 x^{\frac{3}{4}}}{9 x^{\frac{1}{3}}}$$

Factor completely. $$ (x-5)^{-\frac{1}{2}}(x+5)^{-\frac{1}{2}}-(x+5)^{\frac{1}{2}}(x-5)^{-\frac{3}{2}} $$
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