Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 518

\[\boxed{\mathbf{518}.}\]

\[1)\ x^{- 2\sqrt{2}} \bullet \left( \frac{1}{x^{- \sqrt{2} - 1}} \right)^{\sqrt{2} + 1} =\]

\[= x^{- 2\sqrt{2}} \bullet \left( x^{- \left( - \sqrt{2} - 1 \right)} \right)^{\sqrt{2} + 1} =\]

\[= x^{- 2\sqrt{2}} \bullet \left( x^{\sqrt{2} + 1} \right)^{\sqrt{2} + 1} =\]

\[= x^{- 2\sqrt{2}} \bullet x^{\left( \sqrt{2} + 1 \right)^{2}} =\]

\[= x^{- 2\sqrt{2} + \left( \sqrt{2} + 1 \right)^{2}} =\]

\[= x^{- 2\sqrt{2} + \left( \sqrt{2} \right)^{2} + 2\sqrt{2} + 1} =\]

\[= x^{2 + 1} = x^{3};\]

\[2)\ \left( \frac{a^{\sqrt{3}}}{b^{\sqrt{3} - 1}} \right)^{\sqrt{3} + 1} \bullet \frac{a^{- 1 - \sqrt{3}}}{b^{- 2}} =\]

\[= \frac{a^{\sqrt{3} \bullet \left( \sqrt{3} + 1 \right)}}{b^{\left( \sqrt{3} - 1 \right)\left( \sqrt{3} + 1 \right)}} \bullet \frac{b^{- ( - 2)}}{a^{- \left( - 1 - \sqrt{3} \right)}} =\]

\[= \frac{a^{3 + \sqrt{3}}}{b^{\left( \sqrt{3} \right)^{2} - 1}} \bullet \frac{b^{2}}{a^{1 + \sqrt{3}}} =\]

\[= \frac{a^{3 + \sqrt{3}}}{a^{\left( 1 + \sqrt{3} \right)}} \bullet \frac{b^{2}}{b^{3 - 1}} =\]

\[= a^{3 + \sqrt{3} - \left( 1 + \sqrt{3} \right)} \bullet b^{2 - (3 - 1)} =\]

\[= a^{2} \bullet b^{0} = a^{2}.\]