Collection of algebra exercises involving operations with rational expressions.

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

- Dividing by a fraction: multiply by reciprocal:
$$
\frac{t}{s} \cdot \frac{s}{t} = \frac{\cancel{t}}{\cancel{s}} \cdot \frac{\cancel{s}}{\cancel{t}} = 1
$$

✔ Answer: $\boxed{1}$

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

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

✔ Answer: $\boxed{\frac{y}{6}}$

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

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

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

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

✔ Answer: $\boxed{3x + 2}$

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

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

But better to find common denominator.

Let’s write:
$$
\frac{6y - 2}{y} - \frac{6y}{6y + 3}
$$

Factor $6y + 3 = 3(2y + 1)$

Now:
$$
= \frac{6y - 2}{y} - \frac{6y}{3(2y + 1)} = \frac{6y - 2}{y} - \frac{2y}{2y + 1}
$$

Now find LCD: $y(2y + 1)$

$$
= \frac{(6y - 2)(2y + 1)}{y(2y + 1)} - \frac{2y \cdot y}{y(2y + 1)}
$$

Numerator:
$$
(6y - 2)(2y + 1) = 12y^2 + 6y - 4y - 2 = 12y^2 + 2y - 2
$$
Second term: $2y^2$

So:
$$
\frac{12y^2 + 2y - 2 - 2y^2}{y(2y + 1)} = \frac{10y^2 + 2y - 2}{y(2y + 1)}
$$

Factor numerator: $2(5y^2 + y - 1)$ — no clean factorization.

✔ Answer: $\boxed{\frac{10y^2 + 2y - 2}{y(2y + 1)}}$ (or simplified as $\frac{2(5y^2 + y - 1)}{y(2y + 1)}$)

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

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

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

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

✔ 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:
Numerator: $5x^2 + 2x = x(5x + 2)$
Denominator: $4x - 8 = 4(x - 2)$

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

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

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

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

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

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

✔ 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: both numerator and denominator are differences of squares/cubes.

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

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

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

Now:
$$
= \frac{3x + 2}{9x^2 + 6x + 4} \cdot \frac{3x - 2}{3x + 2} = \frac{\cancel{3x + 2}}{9x^2 + 6x + 4} \cdot \frac{3x - 2}{\cancel{3x + 2}} = \frac{3x - 2}{9x^2 + 6x + 4}
$$

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

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

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

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

✔ Answer: $\boxed{\frac{x}{2x - 3}}$

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

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

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

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

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

- Same denominator:
$$
= \frac{5 + x}{x + 2} = \frac{x + 5}{x + 2}
$$

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

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

- Same denominator:
$$
= \frac{9 - 5}{2x + 1} = \frac{4}{2x + 1}
$$

✔ Answer: $\boxed{\frac{4}{2x + 1}}$

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

- Same denominator:
$$
= \frac{17}{r}
$$

✔ Answer: $\boxed{\frac{17}{r}}$

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32. $\frac{x}{z} + \frac{z}{y}$

- Different denominators: LCD = $yz$

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

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

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

- Write 3 as $\frac{3xy}{xy}$:
$$
= \frac{5 - 3xy}{xy}
$$

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

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34. $\frac{8}{m + 3} + \frac{2}{m}$

- LCD: $m(m + 3)$
$$
= \frac{8m}{m(m + 3)} + \frac{2(m + 3)}{m(m + 3)} = \frac{8m + 2m + 6}{m(m + 3)} = \frac{10m + 6}{m(m + 3)}
$$

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

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

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

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

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

- Same denominator:
$$
= \frac{n + 3}{n + 3} = 1
$$

✔ Answer: $\boxed{1}$

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

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

✔ 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. $\frac{3}{x} + \frac{2}{x - 2}$

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

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

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

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

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

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

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

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

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

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

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 Key Techniques Used:

- Factor all expressions before simplifying.
- Use reciprocals for division.
- Combine fractions using common denominators.
- Cancel common factors in numerators and denominators.

Let me know if you'd like these formatted in a table or need help with any specific problem!