Add. Simplify, if possible. $$ \frac{6}{y^{2}+18 y+81}+\frac{3}{y^{2}-81} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{6}{y^{2}+18 y+81} + \frac{3}{y^{2}-81}
$$

**Step 1: Factor the Denominators**

1. **First Denominator:**
   $$
   y^{2} + 18y + 81
   $$
   This is a perfect square trinomial:
   $$
   y^{2} + 18y + 81 = (y + 9)^2
   $$

2. **Second Denominator:**
   $$
   y^{2} - 81
   $$
   This is a difference of squares:
   $$
   y^{2} - 81 = (y - 9)(y + 9)
   $$

**Step 2: Rewrite the Expression with Factored Denominators**
$$
\frac{6}{(y + 9)^2} + \frac{3}{(y - 9)(y + 9)}
$$

**Step 3: Find a Common Denominator**

The least common denominator (LCD) is:
$$
(y + 9)^2 (y - 9)
$$

**Step 4: Adjust Each Fraction to Have the LCD**

1. **First Fraction:**
   Multiply numerator and denominator by $$(y - 9)$$:
   $$
   \frac{6(y - 9)}{(y + 9)^2 (y - 9)}
   $$

2. **Second Fraction:**
   Multiply numerator and denominator by $$(y + 9)$$:
   $$
   \frac{3(y + 9)}{(y + 9)^2 (y - 9)}
   $$

**Step 5: Combine the Fractions**
$$
\frac{6(y - 9) + 3(y + 9)}{(y + 9)^2 (y - 9)}
$$

**Step 6: Simplify the Numerator**
$$
6y - 54 + 3y + 27 = 9y - 27 = 9(y - 3)
$$

**Final Simplified Expression**
$$
\frac{9(y - 3)}{(y - 9)(y + 9)^2}
$$

**Answer:**
$$
\frac{9(y - 3)}{\, (y - 9)(y + 9)^{2} \,}
$$
$$ \frac{9(y - 3)}{(y - 9)(y + 9)^2} $$

Add. Simplify, if possible.