5. Сократите дробь 3 1 a²-a2b-2 1 a2b-1 -a и вычислите ее значение при a = 0,25 и b = . 3

Решение:

Сначала сократим дробь:

\[ \frac{a^3 - a^2b^{-2}}{a^2b^{-1} - a} = \frac{a^2(a - b^{-2})}{a(b^{-1} - 1)} = \frac{a(a - \frac{1}{b^2})}{\frac{1}{b} - 1} = \frac{a \cdot \frac{ab^2 - 1}{b^2}}{\frac{1 - b}{b}} = \frac{a(ab^2 - 1)}{b^2} \cdot \frac{b}{1 - b} = \frac{a(ab^2 - 1)}{b(1 - b)} \]

Теперь подставим значения a = 0,25 = \(\frac{1}{4}\) и b = \(\frac{1}{3}\):

\[ \frac{\frac{1}{4}(\frac{1}{4}(\frac{1}{3})^2 - 1)}{\frac{1}{3}(1 - \frac{1}{3})} = \frac{\frac{1}{4}(\frac{1}{4} \cdot \frac{1}{9} - 1)}{\frac{1}{3} \cdot \frac{2}{3}} = \frac{\frac{1}{4}(\frac{1}{36} - 1)}{\frac{2}{9}} = \frac{\frac{1}{4}(\frac{1 - 36}{36})}{\frac{2}{9}} = \frac{\frac{1}{4} \cdot (-\frac{35}{36})}{\frac{2}{9}} = \frac{-\frac{35}{144}}{\frac{2}{9}} = -\frac{35}{144} \cdot \frac{9}{2} = -\frac{35 \cdot 9}{144 \cdot 2} = -\frac{35}{16 \cdot 2} = -\frac{35}{32} \]

Ответ: -35/32