Simplifying Rational Expressions worksheet with problems in Section A and Section B for algebra practice.

Let’s go through each problem one by one. We’ll simplify the rational expressions by canceling common factors in the numerator and denominator.

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Section A

1)
$\frac{x(x + 3)}{x}$
Cancel $x$ (as long as $x \ne 0$):
→ $x + 3$

2)
$\frac{y(y - 8)}{y}$
Cancel $y$:
→ $y - 8$

3)
$\frac{x(x + 5)}{(x + 5)}$
Cancel $(x + 5)$:
→ $x$

4)
$\frac{8(y - 16)}{4}$
Divide 8 by 4:
→ $2(y - 16)$ or $2y - 32$

5)
$\frac{5(x - 7)}{10x(x - 7)}$
Cancel 5 and 10 → $\frac{1}{2}$, and cancel $(x - 7)$:
→ $\frac{1}{2x}$

6)
$\frac{3x(3x - 4)}{18x^2(3x - 4)}$
Cancel $3x$ from top and bottom:
Numerator: 1
Denominator: $6x$ (since $18x^2 ÷ 3x = 6x$)
Also cancel $(3x - 4)$:
→ $\frac{1}{6x}$

7)
$\frac{x(x + 5)(x - 5)}{(x + 5)}$
Cancel $(x + 5)$:
→ $x(x - 5)$ or $x^2 - 5x$

8)
$\frac{9y(2y - 1)(y - 1)}{27y^2(y - 1)}$
Simplify numbers: $9/27 = 1/3$
Cancel $y$: leaves $y$ in denominator
Cancel $(y - 1)$:
→ $\frac{2y - 1}{3y}$

9)
$\frac{x(x + 1)(x - 1)(x + 1)(x - 1)}{(x + 1)(x - 1)}$
Cancel one $(x + 1)$ and one $(x - 1)$ from top and bottom:
Left with: $x(x + 1)(x - 1)$
Which is $x(x^2 - 1)$ or $x^3 - x$

10)
$\frac{8y(y + 4)^2}{12y^2(y + 4)}$
Simplify numbers: $8/12 = 2/3$
Cancel $y$: leaves $y$ in denominator
Cancel one $(y + 4)$:
→ $\frac{2(y + 4)}{3y}$

11)
$\frac{x(3x - 2)}{7x^3(3x - 2)^2}$
Cancel $x$: leaves $x^2$ in denominator
Cancel one $(3x - 2)$: leaves one in denominator
→ $\frac{1}{7x^2(3x - 2)}$

12)
$\frac{3x^3(5y - 3)(y + 3)}{18x^4(5y - 3)^3}$
Simplify numbers: $3/18 = 1/6$
Cancel $x^3$: leaves $x$ in denominator
Cancel one $(5y - 3)$: leaves $(5y - 3)^2$ in denominator
$(y + 3)$ stays on top
→ $\frac{y + 3}{6x(5y - 3)^2}$

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Section B

1)
$\frac{8x + 4}{2} = \frac{4(2x + 1)}{2} = 2(2x + 1) = 4x + 2$

2)
$\frac{2y + 6}{4} = \frac{2(y + 3)}{4} = \frac{y + 3}{2}$

3)
$\frac{7x}{14x - 21} = \frac{7x}{7(2x - 3)} = \frac{x}{2x - 3}$

4)
$\frac{9y^2}{3y + 27y^2} = \frac{9y^2}{3y(1 + 9y)} = \frac{3y}{1 + 9y}$

5)
$\frac{x - 4}{5x - 20} = \frac{x - 4}{5(x - 4)} = \frac{1}{5}$

6)
$\frac{6y - 30y^2}{24y^2} = \frac{6y(1 - 5y)}{24y^2} = \frac{1 - 5y}{4y}$

7)
$\frac{x - 4}{4 - x} = \frac{-(4 - x)}{4 - x} = -1$

8)
$\frac{21 - 3x}{42x - 6x^2} = \frac{3(7 - x)}{6x(7 - x)} = \frac{3}{6x} = \frac{1}{2x}$

9)
$\frac{x + 2}{x^2 + 7x + 10} = \frac{x + 2}{(x + 2)(x + 5)} = \frac{1}{x + 5}$

10)
$\frac{x^2 - 13x + 36}{x - 4} = \frac{(x - 4)(x - 9)}{x - 4} = x - 9$

11)
$\frac{x^2 - 8x - 20}{9x + 18} = \frac{(x - 10)(x + 2)}{9(x + 2)} = \frac{x - 10}{9}$

12)
$\frac{5x + 40}{x^2 + 6x - 16} = \frac{5(x + 8)}{(x + 8)(x - 2)} = \frac{5}{x - 2}$

13)
$\frac{12x + 20}{9x^2 + 9x - 10}$
Factor numerator: $4(3x + 5)$
Factor denominator: Try factoring $9x^2 + 9x - 10$
Find two numbers that multiply to $9*(-10)=-90$, add to 9 → 15 and -6
So: $9x^2 + 15x - 6x - 10 = 3x(3x + 5) -2(3x + 5) = (3x - 2)(3x + 5)$
So expression becomes: $\frac{4(3x + 5)}{(3x - 2)(3x + 5)} = \frac{4}{3x - 2}$

14)
$\frac{x^2 + 5x + 6}{x^2 + 14x + 24} = \frac{(x + 2)(x + 3)}{(x + 2)(x + 12)} = \frac{x + 3}{x + 12}$

15)
$\frac{x^2 - 7x - 44}{x^2 - 17x + 66}$
Factor numerator: Find two numbers multiplying to -44, adding to -7 → -11 and 4 → $(x - 11)(x + 4)$
Wait: -11 * 4 = -44, but -11 + 4 = -7 → yes!
Denominator: Multiply to 66, add to -17 → -11 and -6 → $(x - 11)(x - 6)$
So: $\frac{(x - 11)(x + 4)}{(x - 11)(x - 6)} = \frac{x + 4}{x - 6}$

16)
$\frac{6x^2 - x - 1}{15x^2 + 8x + 1}$
Factor numerator: 6x² - x - 1 → (3x + 1)(2x - 1)? Let's check: 3x*2x=6x², 3x*(-1)+1*2x = -3x+2x=-x, 1*(-1)=-1 → YES
Denominator: 15x² + 8x + 1 → (5x + 1)(3x + 1)? 5x*3x=15x², 5x*1 + 1*3x=5x+3x=8x, 1*1=1 → YES
So: $\frac{(3x + 1)(2x - 1)}{(5x + 1)(3x + 1)} = \frac{2x - 1}{5x + 1}$

17)
$\frac{x^2 - y^2}{(x + y)^2} = \frac{(x - y)(x + y)}{(x + y)^2} = \frac{x - y}{x + y}$

18)
$\frac{4y^2 - 9x^2}{4x^2y + 6x^3}$
Numerator: difference of squares → $(2y - 3x)(2y + 3x)$
Denominator: factor out $2x^2$ → $2x^2(2y + 3x)$
So: $\frac{(2y - 3x)(2y + 3x)}{2x^2(2y + 3x)} = \frac{2y - 3x}{2x^2}$

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Final Answer:

Section A
1) $x + 3$
2) $y - 8$
3) $x$
4) $2y - 32$
5) $\frac{1}{2x}$
6) $\frac{1}{6x}$
7) $x^2 - 5x$
8) $\frac{2y - 1}{3y}$
9) $x^3 - x$
10) $\frac{2(y + 4)}{3y}$
11) $\frac{1}{7x^2(3x - 2)}$
12) $\frac{y + 3}{6x(5y - 3)^2}$

Section B
1) $4x + 2$
2) $\frac{y + 3}{2}$
3) $\frac{x}{2x - 3}$
4) $\frac{3y}{1 + 9y}$
5) $\frac{1}{5}$
6) $\frac{1 - 5y}{4y}$
7) $-1$
8) $\frac{1}{2x}$
9) $\frac{1}{x + 5}$
10) $x - 9$
11) $\frac{x - 10}{9}$
12) $\frac{5}{x - 2}$
13) $\frac{4}{3x - 2}$
14) $\frac{x + 3}{x + 12}$
15) $\frac{x + 4}{x - 6}$
16) $\frac{2x - 1}{5x + 1}$
17) $\frac{x - y}{x + y}$
18) $\frac{2y - 3x}{2x^2}$