## Task A-3.3. (4 points)

Calculate $144^{\log _{5} 1000}: 10^{6 \log _{5} 12}$.
## Second Solution.

Let's determine the logarithm (base 5) of the desired number:

$$
\begin{aligned}
\log _{5} \frac{144^{\log _{5} 1000}}{10^{6 \log _{5} 12}} & =\log _{5}\left(144^{\log _{5} 1000}\right)-\log _{5}\left(10^{6 \log _{5} 12}\right) \\
& =\log _{5} 1000 \cdot \log _{5} 144-6 \log _{5} 12 \cdot \log _{5} 10 \\
& =\log _{5} 10^{3} \cdot \log _{5} 12^{2}-6 \log _{5} 12 \cdot \log _{5} 10 \\
& =3 \log _{5} 10 \cdot 2 \log _{5} 12-6 \log _{5} 12 \cdot \log _{5} 10=0
\end{aligned}
$$

From this, we get $\frac{144^{\log _{5} 1000}}{10^{6 \log _{5} 12}}=1$.