if $a^{n}$ is irrational then ${a}$ is irrational

Theorem.

If $a$ be a real number and $n$ is an integer such that $a^{n}$ is irrational, then $a$ is irrational.

Proof.

We show this by way of contrapositive. In other words, we show that, if $a$ is rational, then $a^{n}$ is rational.

Let $a$ be rational. Then there exist integers $b$ and $c$ with $c\eq 0$ such that $\\displaystyle a=\\frac{b}{c}$ . Thus, $\\displaystyle a^{n}=\\frac{b^{n}}{c^{n}}$ , which is a rational number.∎

Note that the converse is not true. For example, $\\sqrt{2}$ is irrational and $\\left(\\sqrt{2}\\right)^{2}=2$ is rational.

单词	IfAnIsIrrationalThenaIsIrrational
释义	if $a^{n}$ is irrational then ${a}$ is irrational Theorem. If $a$ be a real number and $n$ is an integer such that $a^{n}$ is irrational, then $a$ is irrational. Proof. We show this by way of contrapositive. In other words, we show that, if $a$ is rational, then $a^{n}$ is rational. Let $a$ be rational. Then there exist integers $b$ and $c$ with $c\eq 0$ such that $\\displaystyle a=\\frac{b}{c}$ . Thus, $\\displaystyle a^{n}=\\frac{b^{n}}{c^{n}}$ , which is a rational number.∎ Note that the converse is not true. For example, $\\sqrt{2}$ is irrational and $\\left(\\sqrt{2}\\right)^{2}=2$ is rational.
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if an is irrational then a is irrational

Theorem.

Proof.

if $a^{n}$ is irrational then ${a}$ is irrational