RATIONAL EXPRESSIONS - Free Printable

It looks like you've uploaded a list of algebraic expressions involving rational expressions (fractions with variables), and the tasks likely involve simplifying, adding, subtracting, multiplying, or dividing these expressions.

Since I can't view images directly, I’ll assume your goal is to solve or simplify each of these rational expressions. Let’s go through them step by step and explain how to approach each type.

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General Rules for Rational Expressions:

1. Multiplication: Multiply numerators and denominators, then simplify.
2. Division: Multiply by the reciprocal.
3. Addition/Subtraction: Find a common denominator, combine numerators, simplify.
4. Factoring: Always factor first before simplifying.

Let’s solve each one:

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13. $\frac{a}{y} \cdot \frac{x}{a}$

- Cancel $a$:
$\frac{\cancel{a}}{y} \cdot \frac{x}{\cancel{a}} = \frac{x}{y}$

✔ Answer: $\boxed{\frac{x}{y}}$

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14. $\frac{t}{s} \div \frac{t}{s}$

- Division: multiply by reciprocal:
$\frac{t}{s} \cdot \frac{s}{t} = \frac{ts}{st} = 1$

✔ Answer: $\boxed{1}$

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15. $\frac{2x}{3y} \cdot \frac{y^2}{4x}$

- Cancel terms:
$\frac{2\cancel{x}}{3\cancel{y}} \cdot \frac{\cancel{y}^2}{4\cancel{x}} = \frac{2y}{3 \cdot 4} = \frac{2y}{12} = \frac{y}{6}$

✔ Answer: $\boxed{\frac{y}{6}}$

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16. $\frac{14b}{9a} \div \frac{7a}{3}$

- Divide: multiply by reciprocal:
$\frac{14b}{9a} \cdot \frac{3}{7a} = \frac{14b \cdot 3}{9a \cdot 7a} = \frac{42b}{63a^2}$

- Simplify: divide numerator and denominator by 21:
$\frac{2b}{3a^2}$

✔ Answer: $\boxed{\frac{2b}{3a^2}}$

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17. $\frac{12(t-1)}{t^2+1} \div \frac{t^2 - 1}{t+1}$

- Note: $t^2 - 1 = (t-1)(t+1)$

So:
$$
\frac{12(t-1)}{t^2+1} \cdot \frac{t+1}{(t-1)(t+1)} = \frac{12(t-1)(t+1)}{(t^2+1)(t-1)(t+1)}
$$

Cancel $(t-1)$ and $(t+1)$:
$$
= \frac{12}{t^2 + 1}
$$

✔ Answer: $\boxed{\frac{12}{t^2 + 1}}$

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18. $\frac{3x^2 + 2x}{x-2} \cdot \frac{x-2}{x}$

- Factor numerator: $3x^2 + 2x = x(3x + 2)$

So:
$$
\frac{x(3x+2)}{x-2} \cdot \frac{x-2}{x} = \frac{x(3x+2)(x-2)}{(x-2)x}
$$

Cancel $x$ and $x-2$:
$$
= 3x + 2
$$

✔ Answer: $\boxed{3x + 2}$

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19. $\frac{6y - 2}{y} - \frac{6y}{6y + 3}$

- First, factor:
- $6y - 2 = 2(3y - 1)$
- $6y + 3 = 3(2y + 1)$

But let's write both terms clearly:
$$
\frac{6y - 2}{y} - \frac{6y}{6y + 3}
$$

We need a common denominator: $y(6y + 3)$

First term: $\frac{(6y - 2)(6y + 3)}{y(6y + 3)}$

Second term: $\frac{6y \cdot y}{y(6y + 3)} = \frac{6y^2}{y(6y + 3)}$

Now subtract:
$$
\frac{(6y - 2)(6y + 3) - 6y^2}{y(6y + 3)}
$$

Expand numerator:
- $(6y - 2)(6y + 3) = 36y^2 + 18y - 12y - 6 = 36y^2 + 6y - 6$
- Subtract $6y^2$: $36y^2 + 6y - 6 - 6y^2 = 30y^2 + 6y - 6$

Factor numerator: $6(5y^2 + y - 1)$

Denominator: $y(6y + 3) = 3y(2y + 1)$

No obvious cancellation → leave as:
$$
\frac{30y^2 + 6y - 6}{y(6y + 3)}
$$

✔ Answer: $\boxed{\frac{30y^2 + 6y - 6}{y(6y + 3)}}$ or simplified: $\boxed{\frac{6(5y^2 + y - 1)}{3y(2y + 1)}} = \boxed{\frac{2(5y^2 + y - 1)}{y(2y + 1)}}$

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20. $\frac{t+2}{t^2 + 3t} \div \frac{t+2}{t+3}$

- First: $t^2 + 3t = t(t + 3)$

So:
$$
\frac{t+2}{t(t+3)} \cdot \frac{t+3}{t+2} = \frac{(t+2)(t+3)}{t(t+3)(t+2)} = \frac{1}{t}
$$

✔ Answer: $\boxed{\frac{1}{t}}$

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21. $\frac{4k - 8}{k + 1} \div \frac{2k - 10}{k + 1}$

- Factor:
- $4k - 8 = 4(k - 2)$
- $2k - 10 = 2(k - 5)$

So:
$$
\frac{4(k - 2)}{k + 1} \cdot \frac{k + 1}{2(k - 5)} = \frac{4(k - 2)(k + 1)}{2(k + 1)(k - 5)}
$$

Cancel $k+1$ and reduce 4/2:
$$
= \frac{2(k - 2)}{k - 5}
$$

✔ Answer: $\boxed{\frac{2(k - 2)}{k - 5}}$

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22. $\frac{5x^2 + 2x}{x - 2} \div \frac{5x + 2}{4x - 8}$

- Factor:
- $5x^2 + 2x = x(5x + 2)$
- $4x - 8 = 4(x - 2)$

So:
$$
\frac{x(5x + 2)}{x - 2} \cdot \frac{4(x - 2)}{5x + 2} = \frac{x(5x+2) \cdot 4(x-2)}{(x-2)(5x+2)} = 4x
$$

✔ Answer: $\boxed{4x}$

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23. $\frac{a^2 + ab}{b} \cdot \frac{b}{a^2 - ab}$

- Factor:
- $a^2 + ab = a(a + b)$
- $a^2 - ab = a(a - b)$

So:
$$
\frac{a(a+b)}{b} \cdot \frac{b}{a(a - b)} = \frac{a(a+b)b}{b \cdot a(a - b)} = \frac{a+b}{a - b}
$$

✔ Answer: $\boxed{\frac{a + b}{a - b}}$

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24. $\frac{10}{x + 2} \cdot \frac{x^2 + 2x}{4x + 6}$

- $x^2 + 2x = x(x + 2)$
- $4x + 6 = 2(2x + 3)$

So:
$$
\frac{10}{x+2} \cdot \frac{x(x+2)}{2(2x+3)} = \frac{10 \cdot x(x+2)}{(x+2) \cdot 2(2x+3)} = \frac{10x}{2(2x+3)} = \frac{5x}{2x+3}
$$

✔ Answer: $\boxed{\frac{5x}{2x+3}}$

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25. $\frac{9x^2 - 4}{27x^3 - 8} \div \frac{3x + 2}{3x - 2}$

- Note: $9x^2 - 4 = (3x)^2 - 2^2 = (3x - 2)(3x + 2)$
- $27x^3 - 8 = (3x)^3 - 2^3 = (3x - 2)(9x^2 + 6x + 4)$

So:
$$
\frac{(3x - 2)(3x + 2)}{(3x - 2)(9x^2 + 6x + 4)} \cdot \frac{3x - 2}{3x + 2}
$$

Simplify:
- Cancel $(3x - 2)$ and $(3x + 2)$ in numerator/denominator:
$$
= \frac{1}{9x^2 + 6x + 4} \cdot (3x - 2)
= \frac{3x - 2}{9x^2 + 6x + 4}
$$

✔ Answer: $\boxed{\frac{3x - 2}{9x^2 + 6x + 4}}$

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26. $\frac{x^2 - x}{4x^2 - 9} \div \frac{x - 1}{2x + 3}$

- $x^2 - x = x(x - 1)$
- $4x^2 - 9 = (2x)^2 - 3^2 = (2x - 3)(2x + 3)$

So:
$$
\frac{x(x - 1)}{(2x - 3)(2x + 3)} \cdot \frac{2x + 3}{x - 1} = \frac{x(x - 1)(2x + 3)}{(2x - 3)(2x + 3)(x - 1)} = \frac{x}{2x - 3}
$$

✔ Answer: $\boxed{\frac{x}{2x - 3}}$

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27. $\frac{x^2 + 10x + 25}{x - 4} \cdot \frac{3x - 12}{2x + 10}$

- $x^2 + 10x + 25 = (x + 5)^2$
- $3x - 12 = 3(x - 4)$
- $2x + 10 = 2(x + 5)$

So:
$$
\frac{(x+5)^2}{x - 4} \cdot \frac{3(x - 4)}{2(x + 5)} = \frac{(x+5)^2 \cdot 3(x - 4)}{(x - 4) \cdot 2(x + 5)} = \frac{3(x+5)}{2}
$$

✔ Answer: $\boxed{\frac{3(x + 5)}{2}}$

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28. $\frac{3}{a} + \frac{2}{b}$

Common denominator: $ab$

$$
= \frac{3b + 2a}{ab}
$$

✔ Answer: $\boxed{\frac{2a + 3b}{ab}}$

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29. $\frac{5}{x + 2} + \frac{x}{x + 2} = \frac{5 + x}{x + 2} = \frac{x + 5}{x + 2}$

✔ Answer: $\boxed{\frac{x + 5}{x + 2}}$

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30. $\frac{9}{2x + 1} - \frac{5}{2x + 1} = \frac{4}{2x + 1}$

✔ Answer: $\boxed{\frac{4}{2x + 1}}$

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31. $\frac{9}{r} + \frac{8}{r} = \frac{17}{r}$

✔ Answer: $\boxed{\frac{17}{r}}$

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32. $\frac{x}{z} + \frac{z}{y}$ — different denominators

Common denominator: $yz$

$$
= \frac{xy + z^2}{yz}
$$

✔ Answer: $\boxed{\frac{xy + z^2}{yz}}$

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33. $\frac{5}{xy} - 3 = \frac{5 - 3xy}{xy}$

✔ Answer: $\boxed{\frac{5 - 3xy}{xy}}$

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34. $\frac{8}{m + 3} + \frac{2}{m} = \frac{8m + 2(m + 3)}{m(m + 3)} = \frac{8m + 2m + 6}{m(m + 3)} = \frac{10m + 6}{m(m + 3)}$

✔ Answer: $\boxed{\frac{10m + 6}{m(m + 3)}}$

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35. $\frac{b}{b - 2} + \frac{4}{b - 3}$

Common denominator: $(b - 2)(b - 3)$

$$
= \frac{b(b - 3) + 4(b - 2)}{(b - 2)(b - 3)} = \frac{b^2 - 3b + 4b - 8}{(b - 2)(b - 3)} = \frac{b^2 + b - 8}{(b - 2)(b - 3)}
$$

✔ Answer: $\boxed{\frac{b^2 + b - 8}{(b - 2)(b - 3)}}$

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36. $\frac{n}{n + 3} + \frac{3}{n + 3} = \frac{n + 3}{n + 3} = 1$

✔ Answer: $\boxed{1}$

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37. $\frac{z}{z + 2} - \frac{4}{z - 2}$

Common denominator: $(z + 2)(z - 2)$

$$
= \frac{z(z - 2) - 4(z + 2)}{(z + 2)(z - 2)} = \frac{z^2 - 2z - 4z - 8}{(z + 2)(z - 2)} = \frac{z^2 - 6z - 8}{z^2 - 4}
$$

✔ Answer: $\boxed{\frac{z^2 - 6z - 8}{z^2 - 4}}$

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38. $\frac{10}{x - y} - \frac{2}{y - x}$

Note: $y - x = -(x - y)$

So:
$$
\frac{10}{x - y} - \frac{2}{-(x - y)} = \frac{10}{x - y} + \frac{2}{x - y} = \frac{12}{x - y}
$$

✔ Answer: $\boxed{\frac{12}{x - y}}$

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39. $\frac{3}{x} + \frac{2}{x - 2}$

Common denominator: $x(x - 2)$

$$
= \frac{3(x - 2) + 2x}{x(x - 2)} = \frac{3x - 6 + 2x}{x(x - 2)} = \frac{5x - 6}{x(x - 2)}
$$

✔ Answer: $\boxed{\frac{5x - 6}{x(x - 2)}}$

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40. $\frac{t}{t + 2} + \frac{2}{t - 1}$

Common denominator: $(t + 2)(t - 1)$

$$
= \frac{t(t - 1) + 2(t + 2)}{(t + 2)(t - 1)} = \frac{t^2 - t + 2t + 4}{(t + 2)(t - 1)} = \frac{t^2 + t + 4}{(t + 2)(t - 1)}
$$

✔ Answer: $\boxed{\frac{t^2 + t + 4}{(t + 2)(t - 1)}}$

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41. $\frac{2x}{x - 1} - \frac{5}{x + 1}$

Common denominator: $(x - 1)(x + 1)$

$$
= \frac{2x(x + 1) - 5(x - 1)}{(x - 1)(x + 1)} = \frac{2x^2 + 2x - 5x + 5}{x^2 - 1} = \frac{2x^2 - 3x + 5}{x^2 - 1}
$$

✔ Answer: $\boxed{\frac{2x^2 - 3x + 5}{x^2 - 1}}$

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42. $\frac{y + 2}{y^2 - 4} + \frac{y^2 + 2y + 4}{y^3 - 8}$

- $y^2 - 4 = (y - 2)(y + 2)$
- $y^3 - 8 = (y - 2)(y^2 + 2y + 4)$

So:
$$
\frac{y + 2}{(y - 2)(y + 2)} + \frac{y^2 + 2y + 4}{(y - 2)(y^2 + 2y + 4)} = \frac{1}{y - 2} + \frac{1}{y - 2} = \frac{2}{y - 2}
$$

✔ Answer: $\boxed{\frac{2}{y - 2}}$

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## ✔ Summary of Answers:

| Problem | Answer |
|--------|--------|
| 13 | $\frac{x}{y}$ |
| 14 | $1$ |
| 15 | $\frac{y}{6}$ |
| 16 | $\frac{2b}{3a^2}$ |
| 17 | $\frac{12}{t^2 + 1}$ |
| 18 | $3x + 2$ |
| 19 | $\frac{2(5y^2 + y - 1)}{y(2y + 1)}$ |
| 20 | $\frac{1}{t}$ |
| 21 | $\frac{2(k - 2)}{k - 5}$ |
| 22 | $4x$ |
| 23 | $\frac{a + b}{a - b}$ |
| 24 | $\frac{5x}{2x + 3}$ |
| 25 | $\frac{3x - 2}{9x^2 + 6x + 4}$ |
| 26 | $\frac{x}{2x - 3}$ |
| 27 | $\frac{3(x + 5)}{2}$ |
| 28 | $\frac{2a + 3b}{ab}$ |
| 29 | $\frac{x + 5}{x + 2}$ |
| 30 | $\frac{4}{2x + 1}$ |
| 31 | $\frac{17}{r}$ |
| 32 | $\frac{xy + z^2}{yz}$ |
| 33 | $\frac{5 - 3xy}{xy}$ |
| 34 | $\frac{10m + 6}{m(m + 3)}$ |
| 35 | $\frac{b^2 + b - 8}{(b - 2)(b - 3)}$ |
| 36 | $1$ |
| 37 | $\frac{z^2 - 6z - 8}{z^2 - 4}$ |
| 38 | $\frac{12}{x - y}$ |
| 39 | $\frac{5x - 6}{x(x - 2)}$ |
| 40 | $\frac{t^2 + t + 4}{(t + 2)(t - 1)}$ |
| 41 | $\frac{2x^2 - 3x + 5}{x^2 - 1}$ |
| 42 | $\frac{2}{y - 2}$ |

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