Exponentiate to its Reciprocal

Algebra Level 2

Is $x$ an irrational number in $\left(\dfrac{a}{b}\right)^{x}=\dfrac{b}{a}$ given that $a \neq b$ and are not zeroes?

No Yes

1 solution

Kaizen Cyrus
May 14, 2019

$\large \begin{aligned} \left( \frac{a}{b} \right)^{x} = & \frac{b}{a} \\ \frac{a^{x}}{b^{x}} = & \frac{b}{a} \end{aligned}$

$x=-1$ is a solution to the equation, which is a rational integer number, not an irrational number.