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12. Выполните умножение \( \frac{6 x^{3}}{x-5} \cdot \frac{25-x^{2}}{18 x^{2}} \)

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Dec 24,2024

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Answer

Результат умножения \( \frac{6 x^{3}}{x-5} \cdot \frac{25-x^{2}}{18 x^{2}} \) равен \( -\frac{5x+x^{2}}{3} \).

Solution

Calculate or simplify the expression \( (6*x^3)/(x-5) * (25-x^2)/(18*x^2) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{6x^{3}}{\left(x-5\right)}\times \left(25-x^{2}\right)}{18x^{2}}\) - step1: Remove the parentheses: \(\frac{\frac{6x^{3}}{x-5}\times \left(25-x^{2}\right)}{18x^{2}}\) - step2: Multiply the terms: \(\frac{-6x^{3}\left(5+x\right)}{18x^{2}}\) - step3: Rewrite the fraction: \(-\frac{6x^{3}\left(5+x\right)}{18x^{2}}\) - step4: Reduce the fraction: \(-\frac{x\left(5+x\right)}{3}\) - step5: Multiply the terms: \(-\frac{5x+x^{2}}{3}\) Умножение \( \frac{6 x^{3}}{x-5} \cdot \frac{25-x^{2}}{18 x^{2}} \) дает \( -\frac{5x+x^{2}}{3} \).

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При умножении дробей, мы умножаем числители и знаменатели. Давайте сначала умножим дроби: \[ \frac{6 x^{3}}{x-5} \cdot \frac{25-x^{2}}{18 x^{2}} = \frac{6 x^{3} \cdot (25-x^{2})}{(x-5) \cdot 18 x^{2}} \] Теперь упростим числитель и знаменатель: Числитель: \(6 x^{3} (25 - x^{2}) = 150 x^{3} - 6 x^{5}\) Знаменатель: \(18 x^{2} (x - 5) = 18 x^{2} (x - 5)\) Теперь мы получаем: \[ \frac{150 x^{3} - 6 x^{5}}{18 x^{2} (x - 5)} \] Чтобы упростить эту дробь, можем вынести общий множитель в числителе: \[ = \frac{6 x^{3} (25 - x)}{18 x^{2} (x - 5)} \] Теперь упростим: \[ = \frac{x (25 - x)}{3 (x - 5)} \] И вот, мы пришли к итоговому результату: \[ \frac{25 - x}{3 (x - 5)} \]

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