Simplify to the same base

Question

$(2^{0.5})^{-2}-(\frac{6^{\sqrt{3}}}{6})^{\sqrt{6}+3}$

I couldn't find a way to simplify to the same base. How do I do that?

I did: $2^{-1}-6^{(\sqrt{3}-1)(\sqrt{6}+3)}$

Answer 1

Suppose we have a number $6^a$, and want to convert it into the form $2^b$. Notice that: $$6^a = 2^{\log_2 6^a} = 2^{a \log_2 6},$$

so when we have $6^{(\sqrt{3}-1)(\sqrt6+3)}$, this becomes $2^{(\sqrt{3}-1)(\sqrt6+3) \log_2{6}}$.

Can you continue?