Simplifying Rational Expressions—Practice Worksheet - Free Printable

To simplify the given rational expressions, we will follow these general steps:

1. Factorize the numerator and denominator where possible.
2. Cancel out common factors between the numerator and denominator.
3. Simplify the remaining expression.

Let's solve each problem step by step.

---

Problem 1:

$$
\frac{8x^2}{12x^4}
$$

- Factor out the greatest common factor (GCF) in both the numerator and denominator:
$$
\frac{8x^2}{12x^4} = \frac{4 \cdot 2 \cdot x^2}{4 \cdot 3 \cdot x^4}
$$
- Cancel out the common factors \(4\) and \(x^2\):
$$
\frac{2}{3x^2}
$$

Answer:
$$
\boxed{\frac{2}{3x^2}}
$$

---

Problem 2:

$$
\frac{24x^2y}{32xy^3}
$$

- Factor out the GCF in both the numerator and denominator:
$$
\frac{24x^2y}{32xy^3} = \frac{8 \cdot 3 \cdot x \cdot x \cdot y}{8 \cdot 4 \cdot x \cdot y \cdot y \cdot y}
$$
- Cancel out the common factors \(8\), \(x\), and \(y\):
$$
\frac{3x}{4y^2}
$$

Answer:
$$
\boxed{\frac{3x}{4y^2}}
$$

---

Problem 3:

$$
\frac{(x+1)^3}{(x+1)^5}
$$

- Use the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\):
$$
\frac{(x+1)^3}{(x+1)^5} = (x+1)^{3-5} = (x+1)^{-2}
$$
- Rewrite using positive exponents:
$$
\frac{1}{(x+1)^2}
$$

Answer:
$$
\boxed{\frac{1}{(x+1)^2}}
$$

---

Problem 4:

$$
\frac{(3x-2)^6}{(3x-2)^4}
$$

- Use the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\):
$$
\frac{(3x-2)^6}{(3x-2)^4} = (3x-2)^{6-4} = (3x-2)^2
$$

Answer:
$$
\boxed{(3x-2)^2}
$$

---

Problem 5:

$$
\frac{2n-3}{3-2n}
$$

- Notice that \(3-2n = -(2n-3)\):
$$
\frac{2n-3}{3-2n} = \frac{2n-3}{-(2n-3)} = -1
$$

Answer:
$$
\boxed{-1}
$$

---

Problem 6:

$$
\frac{15y^2(y+3)}{20y^3(y+3)}
$$

- Cancel out the common factors \(15\), \(y^2\), and \((y+3)\):
$$
\frac{15y^2(y+3)}{20y^3(y+3)} = \frac{15}{20} \cdot \frac{1}{y} = \frac{3}{4y}
$$

Answer:
$$
\boxed{\frac{3}{4y}}
$$

---

Problem 7:

$$
\frac{15y(y+1)}{45y^2(y+1)}
$$

- Cancel out the common factors \(15\), \(y\), and \((y+1)\):
$$
\frac{15y(y+1)}{45y^2(y+1)} = \frac{15}{45} \cdot \frac{1}{y} = \frac{1}{3y}
$$

Answer:
$$
\boxed{\frac{1}{3y}}
$$

---

Problem 8:

$$
\frac{24x^3(4-x)}{18x(4-x)}
$$

- Cancel out the common factors \(6\), \(x\), and \((4-x)\):
$$
\frac{24x^3(4-x)}{18x(4-x)} = \frac{24}{18} \cdot \frac{x^2}{1} = \frac{4x^2}{3}
$$

Answer:
$$
\boxed{\frac{4x^2}{3}}
$$

---

Problem 9:

$$
\frac{6x^3(2-x)}{12x(2-x)}
$$

- Cancel out the common factors \(6\), \(x\), and \((2-x)\):
$$
\frac{6x^3(2-x)}{12x(2-x)} = \frac{6}{12} \cdot \frac{x^2}{1} = \frac{x^2}{2}
$$

Answer:
$$
\boxed{\frac{x^2}{2}}
$$

---

Problem 10:

$$
\frac{12x^3(5-2x)}{18x(2x-5)}
$$

- Notice that \(5-2x = -(2x-5)\):
$$
\frac{12x^3(5-2x)}{18x(2x-5)} = \frac{12x^3(-(2x-5))}{18x(2x-5)} = \frac{-12x^3}{18x} = \frac{-2x^2}{3}
$$

Answer:
$$
\boxed{-\frac{2x^2}{3}}
$$

---

Problem 11:

$$
\frac{a^2 + 6a}{ac + 6c}
$$

- Factorize both the numerator and denominator:
$$
\frac{a^2 + 6a}{ac + 6c} = \frac{a(a + 6)}{c(a + 6)}
$$
- Cancel out the common factor \((a+6)\):
$$
\frac{a}{c}
$$

Answer:
$$
\boxed{\frac{a}{c}}
$$

---

Problem 12:

$$
\frac{x^2 + 4x}{3x + 12}
$$

- Factorize both the numerator and denominator:
$$
\frac{x^2 + 4x}{3x + 12} = \frac{x(x + 4)}{3(x + 4)}
$$
- Cancel out the common factor \((x+4)\):
$$
\frac{x}{3}
$$

Answer:
$$
\boxed{\frac{x}{3}}
$$

---

Problem 13:

$$
\frac{6-8x}{4x^2-3x}
$$

- Factorize both the numerator and denominator:
$$
\frac{6-8x}{4x^2-3x} = \frac{-2(4x-3)}{x(4x-3)}
$$
- Cancel out the common factor \((4x-3)\):
$$
\frac{-2}{x}
$$

Answer:
$$
\boxed{-\frac{2}{x}}
$$

---

Problem 14:

$$
\frac{x^2 + 2x - 15}{x^2 - 10x + 21}
$$

- Factorize both the numerator and denominator:
$$
x^2 + 2x - 15 = (x+5)(x-3)
$$
$$
x^2 - 10x + 21 = (x-7)(x-3)
$$
- The expression becomes:
$$
\frac{(x+5)(x-3)}{(x-7)(x-3)}
$$
- Cancel out the common factor \((x-3)\):
$$
\frac{x+5}{x-7}
$$

Answer:
$$
\boxed{\frac{x+5}{x-7}}
$$

---

Problem 15:

$$
\frac{x^2 - 2x - 3}{x^2 - 5x - 6}
$$

- Factorize both the numerator and denominator:
$$
x^2 - 2x - 3 = (x-3)(x+1)
$$
$$
x^2 - 5x - 6 = (x-6)(x+1)
$$
- The expression becomes:
$$
\frac{(x-3)(x+1)}{(x-6)(x+1)}
$$
- Cancel out the common factor \((x+1)\):
$$
\frac{x-3}{x-6}
$$

Answer:
$$
\boxed{\frac{x-3}{x-6}}
$$

---

Problem 16:

$$
\frac{x^2 - 7x + 12}{x^2 - x - 12}
$$

- Factorize both the numerator and denominator:
$$
x^2 - 7x + 12 = (x-3)(x-4)
$$
$$
x^2 - x - 12 = (x-4)(x+3)
$$
- The expression becomes:
$$
\frac{(x-3)(x-4)}{(x-4)(x+3)}
$$
- Cancel out the common factor \((x-4)\):
$$
\frac{x-3}{x+3}
$$

Answer:
$$
\boxed{\frac{x-3}{x+3}}
$$

---

Problem 17:

$$
\frac{x^2 - 3x - 10}{x^2 - 6x + 5}
$$

- Factorize both the numerator and denominator:
$$
x^2 - 3x - 10 = (x-5)(x+2)
$$
$$
x^2 - 6x + 5 = (x-5)(x-1)
$$
- The expression becomes:
$$
\frac{(x-5)(x+2)}{(x-5)(x-1)}
$$
- Cancel out the common factor \((x-5)\):
$$
\frac{x+2}{x-1}
$$

Answer:
$$
\boxed{\frac{x+2}{x-1}}
$$

---

Problem 18:

$$
\frac{x^2 - 8x + 16}{x^2 - 9x + 20}
$$

- Factorize both the numerator and denominator:
$$
x^2 - 8x + 16 = (x-4)^2
$$
$$
x^2 - 9x + 20 = (x-4)(x-5)
$$
- The expression becomes:
$$
\frac{(x-4)^2}{(x-4)(x-5)}
$$
- Cancel out the common factor \((x-4)\):
$$
\frac{x-4}{x-5}
$$

Answer:
$$
\boxed{\frac{x-4}{x-5}}
$$

---

Problem 19:

$$
\frac{x^2 - 7x + 10}{x^2 - 4x - 5}
$$

- Factorize both the numerator and denominator:
$$
x^2 - 7x + 10 = (x-2)(x-5)
$$
$$
x^2 - 4x - 5 = (x-5)(x+1)
$$
- The expression becomes:
$$
\frac{(x-2)(x-5)}{(x-5)(x+1)}
$$
- Cancel out the common factor \((x-5)\):
$$
\frac{x-2}{x+1}
$$

Answer:
$$
\boxed{\frac{x-2}{x+1}}
$$

---

Problem 20:

$$
\frac{x^2 + 6x + 9}{x^2 + 2x - 3}
$$

- Factorize both the numerator and denominator:
$$
x^2 + 6x + 9 = (x+3)^2
$$
$$
x^2 + 2x - 3 = (x+3)(x-1)
$$
- The expression becomes:
$$
\frac{(x+3)^2}{(x+3)(x-1)}
$$
- Cancel out the common factor \((x+3)\):
$$
\frac{x+3}{x-1}
$$

Answer:
$$
\boxed{\frac{x+3}{x-1}}
$$

---

Problem 21:

$$
\frac{x^2 - 2x - 24}{16 - x^2}
$$

- Factorize both the numerator and denominator:
$$
x^2 - 2x - 24 = (x-6)(x+4)
$$
$$
16 - x^2 = (4-x)(4+x) = -(x-4)(x+4)
$$
- The expression becomes:
$$
\frac{(x-6)(x+4)}{-(x-4)(x+4)}
$$
- Cancel out the common factor \((x+4)\):
$$
\frac{x-6}{-(x-4)} = \frac{-(x-6)}{x-4} = \frac{6-x}{x-4}
$$

Answer:
$$
\boxed{\frac{6-x}{x-4}}
$$

---

Problem 22:

$$
\frac{9 - y^2}{y^2 + 2y - 15}
$$

- Factorize both the numerator and denominator:
$$
9 - y^2 = (3-y)(3+y)
$$
$$
y^2 + 2y - 15 = (y+5)(y-3)
$$
- The expression becomes:
$$
\frac{(3-y)(3+y)}{(y+5)(y-3)}
$$
- Notice that \(3-y = -(y-3)\):
$$
\frac{-(y-3)(3+y)}{(y+5)(y-3)} = \frac{-(3+y)}{y+5}
$$

Answer:
$$
\boxed{-\frac{y+3}{y+5}}
$$

---

Problem 23:

$$
\frac{3x^3 + 18x^2 - 21x}{x^3 + 5x^2 - 14x}
$$

- Factor out the GCF in both the numerator and denominator:
$$
3x^3 + 18x^2 - 21x = 3x(x^2 + 6x - 7) = 3x(x+7)(x-1)
$$
$$
x^3 + 5x^2 - 14x = x(x^2 + 5x - 14) = x(x+7)(x-2)
$$
- The expression becomes:
$$
\frac{3x(x+7)(x-1)}{x(x+7)(x-2)}
$$
- Cancel out the common factors \(x\) and \((x+7)\):
$$
\frac{3(x-1)}{x-2}
$$

Answer:
$$
\boxed{\frac{3(x-1)}{x-2}}
$$

---

Problem 24:

$$
\frac{2x^2 - 18x}{4x^3 - 32x^2 - 36x}
$$

- Factor out the GCF in both the numerator and denominator:
$$
2x^2 - 18x = 2x(x-9)
$$
$$
4x^3 - 32x^2 - 36x = 4x(x^2 - 8x - 9) = 4x(x-9)(x+1)
$$
- The expression becomes:
$$
\frac{2x(x-9)}{4x(x-9)(x+1)}
$$
- Cancel out the common factors \(2x\) and \((x-9)\):
$$
\frac{1}{2(x+1)}
$$

Answer:
$$
\boxed{\frac{1}{2(x+1)}}
$$

---

Problem 25:

$$
\frac{5x^2 - 11x + 2}{5x^2 - 7x - 6}
$$

- Factorize both the numerator and denominator:
$$
5x^2 - 11x + 2 = (5x-1)(x-2)
$$
$$
5x^2 - 7x - 6 = (5x+3)(x-2)
$$
- The expression becomes:
$$
\frac{(5x-1)(x-2)}{(5x+3)(x-2)}
$$
- Cancel out the common factor \((x-2)\):
$$
\frac{5x-1}{5x+3}
$$

Answer:
$$
\boxed{\frac{5x-1}{5x+3}}
$$

---

Problem 26:

$$
\frac{2x^2 - 7x + 3}{2x^2 + 9x - 5}
$$

- Factorize both the numerator and denominator:
$$
2x^2 - 7x + 3 = (2x-1)(x-3)
$$
$$
2x^2 + 9x - 5 = (2x-1)(x+5)
$$
- The expression becomes:
$$
\frac{(2x-1)(x-3)}{(2x-1)(x+5)}
$$
- Cancel out the common factor \((2x-1)\):
$$
\frac{x-3}{x+5}
$$

Answer:
$$
\boxed{\frac{x-3}{x+5}}
$$

---

Problem 27:

$$
\frac{x^2 + 2x - 24}{12 - 4x - x^2}
$$

- Factorize both the numerator and denominator:
$$
x^2 + 2x - 24 = (x+6)(x-4)
$$
$$
12 - 4x - x^2 = -(x^2 + 4x - 12) = -(x+6)(x-2)
$$
- The expression becomes:
$$
\frac{(x+6)(x-4)}{-(x+6)(x-2)}
$$
- Cancel out the common factor \((x+6)\):
$$
\frac{x-4}{-(x-2)} = \frac{-(x-4)}{x-2} = \frac{4-x}{x-2}
$$

Answer:
$$
\boxed{\frac{4-x}{x-2}}
$$

---

Problem 28:

$$
\frac{2x^2 - 8}{x^2 + 4x - 12}
$$

- Factorize both the numerator and denominator:
$$
2x^2 - 8 = 2(x^2 - 4) = 2(x-2)(x+2)
$$
$$
x^2 + 4x - 12 = (x+6)(x-2)
$$
- The expression becomes:
$$
\frac{2(x-2)(x+2)}{(x+6)(x-2)}
$$
- Cancel out the common factor \((x-2)\):
$$
\frac{2(x+2)}{x+6}
$$

Answer:
$$
\boxed{\frac{2(x+2)}{x+6}}
$$

---

Problem 29:

$$
\frac{2x^2 - 9x + 7}{x^2 + 2x - 3}
$$

- Factorize both the numerator and denominator:
$$
2x^2 - 9x + 7 = (2x-7)(x-1)
$$
$$
x^2 + 2x - 3 = (x+3)(x-1)
$$
- The expression becomes:
$$
\frac{(2x-7)(x-1)}{(x+3)(x-1)}
$$
- Cancel out the common factor \((x-1)\):
$$
\frac{2x-7}{x+3}
$$

Answer:
$$
\boxed{\frac{2x-7}{x+3}}
$$

---

Problem 30:

$$
\frac{8x^2 - 14x + 6}{4x - 3}
$$

- Factorize the numerator:
$$
8x^2 - 14x + 6 = 2(4x^2 - 7x + 3) = 2(4x-3)(x-1)
$$
- The expression becomes:
$$
\frac{2(4x-3)(x-1)}{4x-3}
$$
- Cancel out the common factor \((4x-3)\):
$$
2(x-1)
$$

Answer:
$$
\boxed{2(x-1)}
$$

---

Final Answer:

$$
\boxed{
\begin{aligned}
1. & \ \frac{2}{3x^2} \\
2. & \ \frac{3x}{4y^2} \\
3. & \ \frac{1}{(x+1)^2} \\
4. & \ (3x-2)^2 \\
5. & \ -1 \\
6. & \ \frac{3}{4y} \\
7. & \ \frac{1}{3y} \\
8. & \ \frac{4x^2}{3} \\
9. & \ \frac{x^2}{2} \\
10. & \ -\frac{2x^2}{3} \\
11. & \ \frac{a}{c} \\
12. & \ \frac{x}{3} \\
13. & \ -\frac{2}{x} \\
14. & \ \frac{x+5}{x-7} \\
15. & \ \frac{x-3}{x-6} \\
16. & \ \frac{x-3}{x+3} \\
17. & \ \frac{x+2}{x-1} \\
18. & \ \frac{x-4}{x-5} \\
19. & \ \frac{x-2}{x+1} \\
20. & \ \frac{x+3}{x-1} \\
21. & \ \frac{6-x}{x-4} \\
22. & \ -\frac{y+3}{y+5} \\
23. & \ \frac{3(x-1)}{x-2} \\
24. & \ \frac{1}{2(x+1)} \\
25. & \ \frac{5x-1}{5x+3} \\
26. & \ \frac{x-3}{x+5} \\
27. & \ \frac{4-x}{x-2} \\
28. & \ \frac{2(x+2)}{x+6} \\
29. & \ \frac{2x-7}{x+3} \\
30. & \ 2(x-1)
\end{aligned}
}
$$