Simplifying Rational Expressions worksheet with ten algebraic problems for practice.

Let’s go through each problem one by one. We’re simplifying rational expressions — that means we want to cancel out common factors in the numerator and denominator.

---

Problem 1:
Simplify:
  20d⁴ / 70d²

Step 1: Simplify numbers.
20 and 70 both divide by 10 → 20÷10 = 2, 70÷10 = 7 → so 2/7

Step 2: Simplify variables.
d⁴ ÷ d² = d^(4-2) = d²

So answer: 2d² / 7

---

Problem 2:
Simplify:
  27h² / 36h⁴

Step 1: Numbers — 27 and 36. GCF is 9.
27÷9 = 3, 36÷9 = 4 → so 3/4

Step 2: Variables — h² ÷ h⁴ = h^(2-4) = h^(-2) → which is 1/h²

So: 3/(4h²)

Answer: 3 / (4h²)

---

Problem 3:
Simplify:
  24c⁸ / 36c⁹

Step 1: Numbers — 24 and 36. GCF is 12.
24÷12 = 2, 36÷12 = 3 → 2/3

Step 2: Variables — c⁸ ÷ c⁹ = c^(8-9) = c^(-1) = 1/c

So: 2/(3c)

Answer: 2 / (3c)

---

Problem 4:
Simplify:
  24 / (3x - 24)

Step 1: Look at denominator — factor out 3:
3x - 24 = 3(x - 8)

Numerator is 24 → which is 3 × 8

So: 24 / [3(x - 8)] = (3×8) / [3(x - 8)] → cancel 3

Answer: 8 / (x - 8)

---

Problem 5:
Simplify:
  (4p + 36) / 20

Step 1: Factor numerator — 4p + 36 = 4(p + 9)

Denominator: 20 = 4 × 5

So: [4(p + 9)] / (4 × 5) → cancel 4

Answer: (p + 9) / 5

---

Problem 6:
Simplify:
  (6r - 36) / (r - 6)

Step 1: Factor numerator — 6r - 36 = 6(r - 6)

Denominator: (r - 6)

So: [6(r - 6)] / (r - 6) → cancel (r - 6), as long as r ≠ 6

Answer: 6

---

Problem 7:
Simplify:
  (s + 8) / (2s + 16)

Step 1: Factor denominator — 2s + 16 = 2(s + 8)

Numerator: (s + 8)

So: (s + 8) / [2(s + 8)] → cancel (s + 8), as long as s ≠ -8

Answer: 1/2

---

Problem 8:
Simplify:
  (n + 3) / (n² + 4n + 3)

Step 1: Factor denominator — n² + 4n + 3 = (n + 1)(n + 3)

Numerator: (n + 3)

So: (n + 3) / [(n + 1)(n + 3)] → cancel (n + 3), as long as n ≠ -3

Answer: 1 / (n + 1)

---

Problem 9:
Simplify:
  (z² - 8z + 15) / (z - 5)

Step 1: Factor numerator — z² - 8z + 15 = (z - 3)(z - 5)

Denominator: (z - 5)

So: [(z - 3)(z - 5)] / (z - 5) → cancel (z - 5), as long as z ≠ 5

Answer: z - 3

---

Problem 10:
Simplify:
  (g² - 2g - 24) / (g² - g - 20)

Step 1: Factor numerator — g² - 2g - 24
Find two numbers that multiply to -24 and add to -2 → -6 and +4
→ (g - 6)(g + 4)

Step 2: Factor denominator — g² - g - 20
Two numbers that multiply to -20 and add to -1 → -5 and +4
→ (g - 5)(g + 4)

So: [(g - 6)(g + 4)] / [(g - 5)(g + 4)] → cancel (g + 4), as long as g ≠ -4

Answer: (g - 6) / (g - 5)

---

Final Answer:
1) 2d²/7
2) 3/(4h²)
3) 2/(3c)
4) 8/(x - 8)
5) (p + 9)/5
6) 6
7) 1/2
8) 1/(n + 1)
9) z - 3
10) (g - 6)/(g - 5)