## Task B-3.1.

Calculate: $\frac{1}{\left(337^{\frac{1}{2}}+6^{\frac{1}{2}}\right)^{-2}}-\left(\frac{\sqrt{337}-\sqrt{6}}{337^{\frac{3}{2}}-6^{\frac{3}{2}}}\right)^{-1}$.
## Solution.

Let's write the given expression without negative exponents, and express powers with rational exponents using roots (or vice versa). Then we will apply the formulas for the square of a binomial and the difference of cubes. We then get step by step:

$$
\begin{array}{cc}
\frac{1}{\left(337^{\frac{1}{2}}+6^{\frac{1}{2}}\right)^{-2}}-\left(\frac{\sqrt{337}-\sqrt{6}}{337^{\frac{3}{2}}-6^{\frac{3}{2}}}\right)^{-1}=(\sqrt{337}+\sqrt{6})^{2}-\frac{(\sqrt{337})^{3}-(\sqrt{6})^{3}}{\sqrt{337}-\sqrt{6}}= & 2 \text { points } \\
337+2 \sqrt{337} \cdot \sqrt{6}+6-\frac{(\sqrt{337}-\sqrt{6})(337+\sqrt{337} \cdot \sqrt{6}+6)}{\sqrt{337}-\sqrt{6}}= & 2 \text { points } \\
343+2 \sqrt{2022}-(343+\sqrt{2022})=\sqrt{2022} . & 2 \text { points }
\end{array}
$$