Question

In the following exercises, simplify.

$\frac{\frac{3b}{b-5}}{\frac{b^2}{b^2-25}}$

Short answer

The simplified form of the given expression is $\frac{3(b+5)}{b}$.

Step-by-step solution

Step 1. Given information.

The given expression is:

$\frac{\frac{3b}{b-5}}{\frac{b^2}{b^2-25}}$

Step 2. Concepts.

Some concepts are required for simplification:

$\begin{array}{rcl}\frac{A}{B}÷\frac{C}{D}& =& \frac{A}{B}·\frac{D}{C}\\ a^2-b^2& =& (a-b)(a+b)\end{array}$

Step 3. Rewrite the expression and then simplify.

Rewrite the complex fraction as division.

$\frac{3b}{b-5}÷\frac{b^2}{b^2-25}$

Rewrite as the product of first times the reciprocal of the second.

$$\begin{gathered} =\frac{3b}{b-5}·\frac{b^2-25}{b^2} \\ =\frac{3b(b^2-25)}{(b-5)\left(b^2\right)} \\ =\frac{3b(b-5)(b+5)}{(b-5)\left(b\right)\left(b\right)} \\ =\frac{3(b+5)}{b} \end{gathered}$$

Step 4. Conclusion.

The simplified form of the given expression is $\frac{3(b+5)}{b}$.