Questions: Simplify the expression. [ frac8 a^2-35 a b-25 b^28 a b^2+5 b^3 ]

Solution Steps

We start with the numerator \( 8 a^{2} - 35 a b - 25 b^{2} \). Upon factorization, we find: \[ 8 a^{2} - 35 a b - 25 b^{2} = (a - 5 b)(8 a + 5 b) \]

Next, we factor the denominator \( 8 a b^{2} + 5 b^{3} \). The factorization yields: \[ 8 a b^{2} + 5 b^{3} = b^{2}(8 a + 5 b) \]

Now, we can express the original fraction using the factored forms: \[ \frac{8 a^{2} - 35 a b - 25 b^{2}}{8 a b^{2} + 5 b^{3}} = \frac{(a - 5 b)(8 a + 5 b)}{b^{2}(8 a + 5 b)} \] We can cancel the common factor \( (8 a + 5 b) \) from the numerator and the denominator, leading to the simplified expression: \[ \frac{a - 5 b}{b^{2}} \]

Final Answer