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Chapter 3: Problem 84

Using the result that \(\sqrt{2}\) is irrational (proved in Section 0.1), show that \(2^{5 / 2}\) is irrational.

Short Answer

Expert verified
We are given that \(\sqrt{2}\) is irrational and we want to prove that \(2^{5 / 2}\) is irrational as well. We start by simplifying \(2^{5 / 2}\) as \((\sqrt{2})^5\). Next, we assume that \((\sqrt{2})^5\) is rational, which means it can be written as \(\frac{a}{b}\) where \(a\) and \(b\) are integers with no common factors other than 1. Then, we rewrite the expression as \(\sqrt{2} = \frac{a^{\frac{1}{5}}}{b^{\frac{1}{5}}}\). But this contradicts the given information that \(\sqrt{2}\) is irrational, as this form represents a rational number. Based on this contradiction, we conclude that \(2^{5 / 2}\) (or \((\sqrt{2})^5\)) is irrational.

Step by step solution

01

Simplify the expression

First, let's simplify \(2^{5 / 2}\). We can rewrite this as \((2^{1/2})^5\), which is the same as \((\sqrt{2})^5\).
02

Assume the expression is rational

Next, let's assume that \((\sqrt{2})^5\) is rational. This means that \((\sqrt{2})^5\) can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers with no common factors other than 1 (a simplified fractional form).
03

Express \(\sqrt{2}\) in fractional form

Since we assumed that \((\sqrt{2})^5 = \frac{a}{b}\), we can rewrite this expression as \(\sqrt{2} = \frac{a^{\frac{1}{5}}}{b^{\frac{1}{5}}}\).
04

Show a contradiction using the irrationality of \(\sqrt{2}\)

However, we are given that \(\sqrt{2}\) is irrational. This means that \(\sqrt{2}\) cannot be expressed as a fraction in the form \(\frac{a^{\frac{1}{5}}}{b^{\frac{1}{5}}}\), as this form also represents a rational number. This creates a contradiction, since our assumption leads us to conclude that \(\sqrt{2}\) can be written in a rational form.
05

Conclusion

Based on the contradiction found in the previous steps, we must conclude that our assumption was incorrect and \(\sqrt{2}\) cannot be written in the form of \((\sqrt{2})^5 = \frac{a}{b}\). Therefore, we have proved that \(2^{5 / 2}\) (or \((\sqrt{2})^5\)) is irrational.

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