18. $$ \frac{5}{t+4}+\frac{3}{t-4} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{5}{t+4} + \frac{3}{t-4}
$$

**Step 1: Find a Common Denominator**

The denominators are $$ t + 4 $$ and $$ t - 4 $$. The least common denominator (LCD) is:

$$
(t + 4)(t - 4) = t^2 - 16
$$

**Step 2: Rewrite Each Fraction with the LCD**

$$
\frac{5}{t+4} = \frac{5(t - 4)}{(t + 4)(t - 4)} = \frac{5(t - 4)}{t^2 - 16}
$$

$$
\frac{3}{t-4} = \frac{3(t + 4)}{(t - 4)(t + 4)} = \frac{3(t + 4)}{t^2 - 16}
$$

**Step 3: Combine the Fractions**

$$
\frac{5(t - 4) + 3(t + 4)}{t^2 - 16}
$$

**Step 4: Simplify the Numerator**

$$
5(t - 4) + 3(t + 4) = 5t - 20 + 3t + 12 = 8t - 8
$$

**Step 5: Factor the Numerator**

$$
8t - 8 = 8(t - 1)
$$

**Final Simplified Expression**

$$
\frac{8(t - 1)}{t^2 - 16}
$$

or, factored further,

$$
\frac{8(t - 1)}{(t + 4)(t - 4)}
$$

**Summary:**

$$
\frac{5}{t+4} + \frac{3}{t-4} = \frac{8(t - 1)}{t^2 - 16}
$$
$$ \frac{8(t - 1)}{t^2 - 16} $$

18. \( \frac{5}{t+4}+\frac{3}{t-4} \)