III. Write as simple fraction in lowest terms
1.
1
𝑡−1
+
1
𝑡+1
1
𝑡
−
1
𝑡
2
2.
2𝑥
𝑥+1
−𝑎
2
+
𝑎
1
𝑥−𝑎
3. (𝑥+5)
−5−(𝑥+5)
−4
(𝑥+5)−3
4. (𝑥+ℎ)
−1−𝑥
−1
ℎ

Let's simplify each expression as a fraction in its lowest terms.

### 1. 
\[
\frac{1}{t-1} + \frac{1}{t+1} - \frac{1}{t} + \frac{1}{t^2}
\]

**Step 1: Find a common denominator**  
The common denominator for these terms is \( t^2(t-1)(t+1) \).

**Step 2: Express each term with the common denominator**
\[
\frac{1}{t-1} = \frac{t^2(t+1)}{t^2(t-1)(t+1)}
\]
\[
\frac{1}{t+1} = \frac{t^2(t-1)}{t^2(t-1)(t+1)}
\]
\[
\frac{1}{t} = \frac{t(t-1)(t+1)}{t^2(t-1)(t+1)}
\]
\[
\frac{1}{t^2} = \frac{(t-1)(t+1)}{t^2(t-1)(t+1)}
\]

**Step 3: Combine the fractions**
\[
\frac{t^2(t+1) + t^2(t-1) - t(t-1)(t+1) + (t-1)(t+1)}{t^2(t-1)(t+1)}
\]

**Step 4: Simplify the numerator**
Simplify the expression in the numerator, then cancel out any common factors with the denominator.

### 2.
\[
\frac{2x}{x+1} - a^2 + \frac{a}{x-a}
\]

**Step 1: Find a common denominator**  
The common denominator is \((x+1)(x-a)\).

**Step 2: Express each term with the common denominator**
\[
\frac{2x(x-a)}{(x+1)(x-a)} - \frac{a^2(x+1)(x-a)}{(x+1)(x-a)} + \frac{a(x+1)}{(x+1)(x-a)}
\]

**Step 3: Combine the fractions**
\[
\frac{2x(x-a) - a^2(x+1)(x-a) + a(x+1)}{(x+1)(x-a)}
\]

**Step 4: Simplify the numerator**
Simplify the expression in the numerator.

### 3.
\[
\frac{(x+5) - 5 - (x+5) - 4}{(x+5) - 3}
\]

**Step 1: Simplify the numerator**
Combine like terms in the numerator:
\[
(x+5) - 5 - (x+5) - 4 = x + 5 - 5 - x - 5 - 4 = -4
\]

**Step 2: Simplify the expression**
The fraction becomes:
\[
\frac{-4}{x+2}
\]

### 4.
\[
\frac{(x+h) - 1 - x - 1}{h}
\]

**Step 1: Simplify the numerator**
Combine like terms in the numerator:
\[
(x+h) - 1 - x - 1 = h - 2
\]

**Step 2: Simplify the expression**
The fraction becomes:
\[
\frac{h - 2}{h}
\]

### Final Simplified Expressions
1. \(\frac{t^2(t+1) + t^2(t-1) - t(t-1)(t+1) + (t-1)(t+1)}{t^2(t-1)(t+1)}\) (Further simplification might be possible)
2. \(\frac{2x(x-a) - a^2(x+1)(x-a) + a(x+1)}{(x+1)(x-a)}\) (Further simplification might be possible)
3. \(\frac{-4}{x+2}\)
4. \(\frac{h - 2}{h}\)

Would you like to see the detailed simplification of any of these expressions or have any questions?

---

### Related Questions:
1. How do you find a common denominator for rational expressions?
2. What are some methods for simplifying complex fractions?
3. How do you factor polynomials to simplify fractions?
4. Can you explain the distributive property in simplifying algebraic expressions?
5. What is the significance of lowest terms in a fraction?

**Tip:** When simplifying algebraic fractions, always factorize the numerator and the denominator where possible before canceling out common factors.

III. Write as simple fraction in lowest terms
1.
1/(t-1) + 1/(t+1) - 1/t + 1/t^2
2. 2x/(x+1) - a^2 + a/(x-a)
3. (x+5) - 5 - (x+5) - 4/(x+5) - 3
4. (x+h) - 1 - x - 1/h

This problem demonstrates how to simplify rational expressions to their lowest terms. The solution covers key algebraic concepts, including finding a common denominator, combining fractions, and simplifying expressions. Suitable for students in Grades 9-11.

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