## Task B-1.1.

Write the expression $\left[27^{-2 m+3} \cdot\left(\frac{1}{9}\right)^{3-2 m}\right]^{-2}: 81^{1+m}-6 \cdot 81^{-3}$, where $m$ is an integer, in the form of a power with a positive exponent.
## Solution.

Let's simplify the given expression:

$$
\begin{array}{ll}
{\left[27^{-2 m+3} \cdot\left(\frac{1}{9}\right)^{3-2 m}\right]^{-2}: 81^{1+m}-6 \cdot 81^{-3}=} & 2 \text { points } \\
=\left[3^{9-6 m} \cdot 3^{4 m-6}\right]^{-2}: 3^{4+4 m}-6 \cdot 3^{-12}= & 1 \text { point } \\
=\left[3^{3-2 m}\right]^{-2}: 3^{4+4 m}-6 \cdot 3^{-12}= & \\
=3^{4 m-6}: 3^{4+4 m}-6 \cdot 3^{-12}= & 1 \text{point} \\
=3^{-10}-6 \cdot 3^{-12}= & 1 \text{point} \\
=3^{-12} \cdot\left(3^{2}-6\right)=3 \cdot 3^{-12}=3^{-11} & 1 \text{point}
\end{array}
$$