(38) $$ \frac{\frac{2 r+10}{r^{2}}}{\frac{r^{2}-25}{4 r}} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{\frac{2r + 10}{r^2}}{\frac{r^2 - 25}{4r}}
$$

**Step 1: Rewrite the Division as Multiplication**

$$
\frac{2r + 10}{r^2} \div \frac{r^2 - 25}{4r} = \frac{2r + 10}{r^2} \times \frac{4r}{r^2 - 25}
$$

**Step 2: Factor Where Possible**

- Factor $$2r + 10$$:
  
  $$
  2r + 10 = 2(r + 5)
  $$

- Factor $$r^2 - 25$$ (difference of squares):

$$
  r^2 - 25 = (r + 5)(r - 5)
  $$

**Step 3: Substitute the Factors Back Into the Expression**

$$
\frac{2(r + 5)}{r^2} \times \frac{4r}{(r + 5)(r - 5)}
$$

**Step 4: Cancel Common Factors**

- The $$(r + 5)$$ terms cancel out:

$$
\frac{2}{r^2} \times \frac{4r}{r - 5}
$$

**Step 5: Multiply the Remaining Terms**

$$
\frac{2 \times 4r}{r^2 (r - 5)} = \frac{8r}{r^2 (r - 5)}
$$

**Step 6: Simplify the Expression**

$$
\frac{8r}{r^2 (r - 5)} = \frac{8}{r (r - 5)}
$$

**Final Simplified Form:**

$$
\boxed{\dfrac{8}{\,r(r-5)\,}}
$$
$$ \frac{8}{r(r - 5)} $$

(38) \( \frac{\frac{2 r+10}{r^{2}}}{\frac{r^{2}-25}{4 r}} \)