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

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