24 lines
837 B
TeX
24 lines
837 B
TeX
\beginsection{2.2}
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Show that $2^{1/3}$, $5^{1/7}$, and $(13)^{1/4}$ do not represent
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rational numbers.
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\medskip
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$2^{1/3}$ is a solution to the polynomial $x^3-2=0$.
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By the Rational Zeroes Theorem, the only possible rational solutions are
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$\pm1$ and $\pm2$, neither of which is an actual solution.
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Therefore, $2^{1/3}$ is not a rational number.
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\medskip
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$5^{1/7}$ is a solution to the polynomial $x^7-5=0$.
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By the Rational Zeroes Theorem, the only possible rational solutions are
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$\pm1$ and $\pm5$, neither of which is an actual solution.
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Therefore, $5^{1/7}$ is not a rational number.
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\medskip
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$13^{1/4}$ is a solution to the polynomial $x^4-13=0$.
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By the Rational Zeroes Theorem, the only possible rational solutions are
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$\pm1$ and $\pm13$, neither of which is an actual a solution.
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Therefore, $13^{1/4}$ is not a rational number.
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