تصحيح تمارين القوى في الأعداد الكسرية
\[ \left( \frac{5}{7} \right)^{-120} \]
\[
\left( \frac{5}{7} \right)^{-120} = \left( \frac{7}{5} \right)^{120}
\]
\[ (-0,9)^{-2009} \times \left( \frac{307}{333} \right)^{-2009} \]
\[
\begin{aligned}
& (-0,9)^{-2009} \times \left( \frac{307}{333} \right)^{-2009} \\
= & \left( -0,9 \times \frac{307}{333} \right)^{-2009} \\
= & \left( -\frac{9}{10} \times \frac{307}{333} \right)^{-2009} \\
= & \left( -\frac{307}{370} \right)^{-2009} \\
= & \left( -\frac{370}{307} \right)^{2009}
\end{aligned}
\]
\[ (-6,17)^{147} \times (-6,17)^{23} \]
\[
(-6,17)^{147} \times (-6,17)^{23} = (-6,17)^{147 + 23} = (-6,17)^{170}
\]
\[ -(2^3)^5 \]
\[
-(2^3)^5 = -2^{3 \times 5} = -2^{15}
\]
\[ \frac{(-3,14)^{-219}}{(3,14)^{124}} \times \frac{1}{(-3)^{-343}} \]
\[
\begin{aligned}
& \frac{(-3,14)^{-219}}{(3,14)^{124}} \times \frac{1}{(-3)^{-343}} \\
= & (-3,14)^{-219} \times (3,14)^{-124} \times (-3)^{343} \\
= & (-1)^{-219} \times (3,14)^{-219} \times (3,14)^{-124} \times (-3)^{343} \\
= & -1 \times (3,14)^{-343} \times (-1)^{343} \times 3^{343} \\
= & - (3,14)^{-343} \times (-1) \times 3^{343} \\
= & (3,14)^{-343} \times 3^{343} \\
= & \left( \frac{3}{3,14} \right)^{343}
\end{aligned}
\]
\[ \frac{(-19)^{167}}{(-19)^{57}} \]
\[
\frac{(-19)^{167}}{(-19)^{57}} = (-19)^{167 - 57} = (-19)^{110}
\]
\[ \left( \frac{3}{5} \right)^{-2} \]
\[
\left( \frac{3}{5} \right)^{-2} = \left( \frac{5}{3} \right)^2 = \frac{25}{9}
\]
\[ \left( -\frac{1}{2} \right)^3 \]
\[
\left( -\frac{1}{2} \right)^3 = -\frac{1}{8}
\]
\[ \left( \frac{20}{17} \right)^{2017} \times \left( \frac{20}{17} \right)^{-2016} \]
\[
\left( \frac{20}{17} \right)^{2017} \times \left( \frac{20}{17} \right)^{-2016} = \left( \frac{20}{17} \right)^{2017 - 2016} = \frac{20}{17}
\]
\[ \left[ \left( \frac{20}{17} \right)^{-2354} \right]^0 \]
\[
\left[ \left( \frac{20}{17} \right)^{-2354} \right]^0 = 1
\]