Question: Simplify each rational algebraic expression.
Apply factoring when necessary.
1. $$\frac{21x^{2}y}{7xy^{2}}$$
2. $$\frac{9a^{3}b^{2}}{27a^{2}b}$$
3. $$\frac{7x^{2}}{14x - 21}$$
4. $$\frac{15n + 30}{5n + 10}$$
5. $$\frac{9x^{2} - 27x}{3x - 9}$$
1. **Simplify** $$\frac{21x^{2}y}{7xy^{2}}$$
- Factor numerator and denominator:
$$21x^{2}y = 3 \times 7 \times x^{2} \times y$$
$$7xy^{2} = 7 \times x \times y^{2}$$
- Cancel common factors 7, x, and y:
$$\frac{\cancel{7} x^{\cancel{2}} y^{\cancel{1}} 3}{\cancel{7} x y^{\cancel{2}}} = \frac{3x}{y}$$
2. **Simplify** $$\frac{9a^{3}b^{2}}{27a^{2}b}$$
- Factor numerator and denominator:
$$9a^{3}b^{2} = 9 \times a^{3} \times b^{2}$$
$$27a^{2}b = 27 \times a^{2} \times b$$
- Cancel common factors 9, $$a^{2}$$, and b:
$$\frac{\cancel{9} a^{\cancel{2}} a b^{\cancel{1}} b}{3 \times \cancel{9} a^{\cancel{2}} b} = \frac{a b}{3}$$
3. **Simplify** $$\frac{7x^{2}}{14x - 21}$$
- Factor denominator:
$$14x - 21 = 7(2x - 3)$$
- Write expression:
$$\frac{7x^{2}}{7(2x - 3)}$$
- Cancel common factor 7:
$$\frac{\cancel{7} x^{2}}{\cancel{7} (2x - 3)} = \frac{x^{2}}{2x - 3}$$
4. **Simplify** $$\frac{15n + 30}{5n + 10}$$
- Factor numerator and denominator:
$$15n + 30 = 15(n + 2)$$
$$5n + 10 = 5(n + 2)$$
- Cancel common factor $$n + 2$$:
$$\frac{15 \cancel{(n + 2)}}{5 \cancel{(n + 2)}} = \frac{15}{5} = 3$$
5. **Simplify** $$\frac{9x^{2} - 27x}{3x - 9}$$
- Factor numerator and denominator:
$$9x^{2} - 27x = 9x(x - 3)$$
$$3x - 9 = 3(x - 3)$$
- Cancel common factor $$x - 3$$:
$$\frac{9x \cancel{(x - 3)}}{3 \cancel{(x - 3)}} = \frac{9x}{3}$$
- Simplify fraction:
$$\frac{\cancel{3} 3x}{\cancel{3} 1} = 3x$$
**Final answers:**
1. $$\frac{3x}{y}$$
2. $$\frac{a b}{3}$$
3. $$\frac{x^{2}}{2x - 3}$$
4. $$3$$
5. $$3x$$