<s>[INST] Solve power with negative exponent $\frac{125^{6}\times 25^{-3}}{(5^{2})^{-3}\times25^{7}}$ I am currently studying about exponents and powers for college calculus discipline. In the meantime I came across negative exponents, like this $25^{-3}$ and $(5^{2})^{-3}$.
I have this calculation to solve,
$$
\frac{125^{6}\times 25^{-3}}{(5^{2})^{-3}\times25^{7}}
$$
But, I get very confused and I end up getting stuck in calculations with negative exponents (i.e. $b^{-a}$) and I do not know how to solve them. So I'd like to know. How can I resolve powers with negative exponents?
 [/INST] \begin{align}
\frac{125^6\times 25^{-3}}{(5^2)^{-3}\times 25^7}=
\frac{\left(\dfrac{125^6}{25^3}\right)}{\left(\dfrac{25^7}{(5^2)^3}\right)}=
\dfrac{125^6}{25^3}\dfrac{(5^2)^3}{25^7}=
\dfrac{(5^3)^6}{(5^2)^3}\dfrac{(5^2)^3}{(5^2)^7}=
\dfrac{5^{3\times 6 + 2\times 3}}{5^{2\times 3 + 2\times 7}}=
5^{24-20}=
5^4=
625.
\end{align}
</s>