تمارين في القوى - الهدية المفصلة
A) \[E = \left( \frac{-18}{8} \right)^2 \times \left( \frac{9}{4} \right)^{-5}\]
\[\left( \frac{-18}{8} \right)^2 = \left( \frac{-9 \cdot 2}{4 \cdot 2} \right)^2 = \left( \frac{-9}{4} \right)^2 = \frac{81}{16} \\ \left( \frac{9}{4} \right)^{-5} = \left( \frac{3^2}{2^2} \right)^{-5} = \frac{1}{\left( \frac{9}{4} \right)^5} = \frac{1}{\frac{59049}{1024}} \\ \text{جداء قوّتين لعددين كسريّين}\]
B) \[D = \left( \frac{-3}{5} \right)^7 \times \left( \frac{-5}{3} \right)^3\]
\[\left( \frac{-3}{5} \right)^7 \cdot \left( \frac{-5}{3} \right)^3 = \left( \frac{-3}{5} \cdot \frac{-5}{3} \right)^3 \cdot \left( \frac{-3}{5} \right)^{4} = 1^3 \cdot \left( \frac{-3}{5} \right)^4 = \left( \frac{-3}{5} \right)^4\]
C) \[G = \frac{\left( \frac{3}{4} \right)^5 \cdot \left( \frac{5}{2} \right)^{12}}{\left( \frac{-5}{2} \right)^{15} \cdot \left( \frac{-3}{4} \right)^4}\]
\[\text{البسط :} \left( \frac{3}{4} \right)^5 \cdot \left( \frac{5}{2} \right)^{12} = \frac{3^5 \cdot 5^{12}}{4^5 \cdot 2^{12}} \\ \text{المقام :} \left( \frac{-5}{2} \right)^{15} \cdot \left( \frac{-3}{4} \right)^4 = \frac{5^{15} \cdot 3^4}{2^{15} \cdot 4^4} \\ \text{إذن} G = \frac{3^5 \cdot 5^{12}}{4^5 \cdot 2^{12}} \cdot \frac{2^{15} \cdot 4^4}{5^{15} \cdot 3^4} = \frac{3^{5-4} \cdot 2^{15-12} \cdot 4^4}{4^5 \cdot 5^{15-12}}\]
D) \[F = \left( \frac{-3}{5} \right)^5 \times \left( \frac{3}{5} \right)^{-4}\]
\[\left( \frac{-3}{5} \right)^5 \cdot \left( \frac{3}{5} \right)^{-4} = \left( \frac{-1 \cdot 3^5}{5^5} \cdot \frac{5^4}{3^4} \right) = \left( -1 \cdot \frac{3^{5-4}}{5^{5-4}} \right) = -\frac{3}{5}\]
E) \[I = \frac{\left( \frac{5}{3} \right)^{12}}{\left[ (-10)^{-4} \right]^{-3}}\]
\[I = \frac{\left( \frac{5}{3} \right)^{12}}{\left[ (-10)^{-4} \right]^{-3}} = \frac{\left( \frac{5}{3} \right)^{12}}{(-10)^{12}} = \frac{5^{12}}{3^{12} \cdot (-10)^{12}} = \frac{5^{12}}{3^{12} \cdot 10^{12}} = \frac{1}{3^{12}}\]
F) \[H = \frac{(10^2)^3 \cdot 10^{-5}}{(10^{-1})^4 \cdot (10^{-2})^{-3}}\]
\[(10^2)^3 = 10^{6} \\ \text{البسط :} 10^6 \cdot 10^{-5} = 10^{6 - 5} = 10^1 \\ \text{المقام :} (10^{-1})^4 = 10^{-4}, (10^{-2})^{-3} = 10^6 \\ \text{المجموع :} \frac{10^1}{10^{-4} \cdot 10^6} = \frac{10^1}{10^{2}} = 10^{-1}\]