How to show that $\lim_{n\rightarrow \infty }{a_n}^{b_n}=\alpha ^\beta $?

Notice that $$a_n^{b_n}=e^{b_n\log(a_n)}$$ so by the continuity of the exponential and logarithmic functions you have the result, of course with the assumption $\boldsymbol{a_n>0}$ and $\boldsymbol{\alpha>0}$.

I think that we can use the fact of convergent sequences product and that $$e^{b_n \cdot \log a_n}$$ converges when $n \rightarrow \infty$.