Problem with generalized likelihood ratio test from samples from beta distribution

Question

I was trying to resolve this exercise:

This exercise is from the book "Statistical Inference, Second Edition" by Casella and Berger. Checking the solutions manual, I was understanding the solution but I can't figure out why the operation that is hightlighted in yelow it's equal to one. My possition is that the value hightlighted depends on the sample, but the solution is hardly wrong. Can someone explain that part?. Here's the solution:

Answer 1

Set everything into an exponential:

\begin{align*} \left(\prod_{i=1}^nx_i\right)^{\hat\theta_0-\hat\mu}\left(\prod_{i=1}^my_i\right)^{\hat\theta_0-\hat\theta}&=\exp\left\{ (\hat\theta_0-\hat\mu)\sum_i\log(x_i)+(\hat\theta_0-\hat\theta)\sum_i\log(y_i)\right\}\\ &=\exp\left\{ \dfrac{-(n+m)\sum_i\log(x_i)}{\sum_i\log(x_i)+\sum_i\log(y_i)}+n+\dfrac{-(n+m)\sum_i\log(y_i)}{\sum_i\log(x_i)+\sum_i\log(y_i)}+m\right\}\\ &=\exp\left\{ \dfrac{-(n+m)\left[\sum_i\log(x_i)+\sum_i\log(y_i)\right]}{\sum_i\log(x_i)+\sum_i\log(y_i)}+n+m\right\}\\ &=\exp\left\{-(n+m)+n+m\right\}\\ &=1 \end{align*}