Questions: Find the quotient. Simplify your answer. (b-7)/b^2 ÷ (b^2-49)/6b

Solution Steps

To find the quotient of two fractions, we multiply the first fraction by the reciprocal of the second fraction. After that, we simplify the resulting expression by factoring and canceling common terms.

We need to find the quotient of the division of two rational expressions: \[ \frac{b-7}{b^{2}} \div \frac{b^{2}-49}{6b} \]

To divide by a fraction, we multiply by its reciprocal. Thus, the expression becomes: \[ \frac{b-7}{b^{2}} \times \frac{6b}{b^{2}-49} \]

First, factor the expression \(b^2 - 49\) as a difference of squares: \[ b^2 - 49 = (b - 7)(b + 7) \] Substitute this into the expression: \[ \frac{b-7}{b^{2}} \times \frac{6b}{(b-7)(b+7)} \]

Cancel the common factor \((b-7)\) from the numerator and the denominator: \[ \frac{1}{b^{2}} \times \frac{6b}{b+7} = \frac{6b}{b^{2}(b+7)} \]

Simplify the expression by canceling \(b\) from the numerator and one \(b\) from the denominator: \[ \frac{6}{b(b+7)} \]

Final Answer