# Prove or disprove that if a and b are rational numbers, then a^{b} is also

Prove or disprove that if a and b are rational numbers, then ${a}^{b}$ is also rational.
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Given statement: If a and b are rational numbers, then ${a}^{b}$ is also rational.
The given statement is false, which will be disproven using a counterexample.
Counterexample
If $a=-1$ and $b=\frac{1}{2}$, we then obtain ${\left(-1\right)}^{\frac{1}{2}}=\sqrt{-1}$ which is not rational number (it is called an imaginary number) and thus ${a}^{b}$ is not necessarily rational.
Result:
Statement is false, one counterexample is for a=-1 and $b=\frac{1}{2}$.