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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 means to get rid of any square roots in the denominator. For example, \(\frac{1}{\sqrt{5}}\) can be rationalized to \(\frac{\sqrt{5}}{5}\) and \(\frac{1}{5+\sqrt{5}}\) can be rationalized to \(\frac{5 - \sqrt{5}}{20}\).

Step by step solution

01

Rationalize the denominator of the first example

In the first example \(\frac{1}{\sqrt{5}}\), rationalizing the denominator involves multiplying both the numerator and the denominator by \(\sqrt{5}\). This gives: \(\frac{1* \sqrt{5}}{\sqrt{5}*\sqrt{5}}\). Simplifying this results in \(\frac{\sqrt{5}}{5}\).
02

Rationalize the denominator of the second example

In the second example \(\frac{1}{5 + \sqrt{5}}\), to rationalize the denominator, use the method of multiplying by the conjugate. The conjugate of the denominator \(5 + \sqrt{5}\) is \(5 - \sqrt{5}\). Therefore, multiply both, the numerator and the denominator, by \(5 - \sqrt{5}\). This results in: \(\frac{1*(5 - \sqrt{5})}{(5 + \sqrt{5})*(5 - \sqrt{5})}\). This simplifies to \(\frac{5 - \sqrt{5}}{5^2 - (\sqrt{5})^2}\), further simplifying to \(\frac{5 - \sqrt{5}}{20}\).
03

Final explanation

Rationalizing a denominator basically means transforming it into a rational number if it currently contains an irrational number. It is achieved by multiplying by an appropriate form of 1 (which is a number divided by itself), leading to the elimination of the square root in the denominator. It is a common operation in algebra, particularly when working with irrational roots.

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Most popular questions from this chapter

It takes you 50 minutes to get to campus. You spend t minutes walking to the bus stop and the rest of the time riding the bus. Your walking rate is 0.06 mile per minute and the bus travels at a rate of 0.5 mile per minute. The total distance walking and traveling by bus is given by the algebraic expression $$0.06 t+0.5(50-t)$$ a. Simplify the algebraic expression. b. Use each form of the algebraic expression to determine the total distance that you travel if you spend 20 minutes walking to the bus stop.

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The early Greeks believed that the most pleasing of all rectangles were golden rectangles, whose ratio of width to height is $$\frac{w}{h}=\frac{2}{\sqrt{5}-1}$$ The Parthenon at Athens fits into a golden rectangle once the triangular pediment is reconstructed. (IMAGE CANNOT COPY) Rationalize the denominator of the golden ratio. Then use a calculator and find the ratio of width to height, correct to the nearest hundredth, in golden rectangles.

Using an example, explain how to factor out the greatest common factor of a polynomial.

Simplify by reducing the index of the radical. $$ \sqrt[6]{x^{4}} $$
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