Question: Simplify the following rational expression and
express in expanded form.
$$\frac{k^{10} + 9k^{5} + 18}{k^{10} + 6k^{5} + 9} = \frac{k^{5} + 6}{k^{5} + 3}$$
1. **State the problem:** Simplify the rational expression $$\frac{k^{10} + 9k^{5} + 18}{k^{10} + 6k^{5} + 9}$$ and express it in expanded form.
2. **Rewrite the expression:** We want to simplify $$\frac{k^{10} + 9k^{5} + 18}{k^{10} + 6k^{5} + 9}$$.
3. **Recognize powers:** Notice that $$k^{10} = (k^{5})^{2}$$, so let $$x = k^{5}$$ to simplify the expression.
4. **Rewrite numerator and denominator in terms of $$x$$:**
Numerator: $$x^{2} + 9x + 18$$
Denominator: $$x^{2} + 6x + 9$$
5. **Factor numerator and denominator:**
- Numerator: Find factors of 18 that add to 9, which are 6 and 3.
$$x^{2} + 9x + 18 = (x + 6)(x + 3)$$
- Denominator: Recognize perfect square trinomial:
$$x^{2} + 6x + 9 = (x + 3)^{2}$$
6. **Rewrite the fraction with factors:**
$$\frac{(x + 6)(x + 3)}{(x + 3)^{2}}$$
7. **Cancel common factors:**
$$\frac{\cancel{(x + 3)}(x + 6)}{\cancel{(x + 3)}(x + 3)} = \frac{x + 6}{x + 3}$$
8. **Substitute back $$x = k^{5}$$:**
$$\frac{k^{5} + 6}{k^{5} + 3}$$
9. **Express in expanded form:** The expression is already simplified and expanded as $$\frac{k^{5} + 6}{k^{5} + 3}$$.
**Final answer:** $$\frac{k^{5} + 6}{k^{5} + 3}$$