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The image shows four mathematical expressions that require simplification. Let's go through each one step by step and simplify them.

### (1) Simplify \( \frac{7x^3}{x^4} \)

We can simplify this by dividing the powers of \(x\). Using the rule \( \frac{x^a}{x^b} = x^{a-b} \), we get:

\[
\frac{7x^3}{x^4} = 7x^{3-4} = 7x^{-1} = \frac{7}{x}
\]

Thus, the simplified form is:

\[
\frac{7}{x}
\]

---

### (2) Simplify \( \frac{15a^6b}{6a^2b^2} \)

We can simplify the coefficients and the powers of \(a\) and \(b\). Start by simplifying the coefficients:

\[
\frac{15}{6} = \frac{5}{2}
\]

Now, simplify the powers of \(a\) and \(b\):

\[
\frac{a^6}{a^2} = a^{6-2} = a^4 \quad \text{and} \quad \frac{b}{b^2} = b^{1-2} = b^{-1} = \frac{1}{b}
\]

Thus, the simplified form is:

\[
\frac{5a^4}{2b}
\]

---

### (3) Simplify \( \frac{2(x-2)}{(x-2)(x+3)} \)

We can cancel the common factor \( (x - 2) \) from the numerator and denominator:

\[
\frac{2(x-2)}{(x-2)(x+3)} = \frac{2}{x+3}, \quad \text{where} \, x \neq 2 \, (\text{to avoid division by zero})
\]

Thus, the simplified form is:

\[
\frac{2}{x+3}
\]

---

### (4) Simplify \( \frac{(x-1)(x+3)}{(x+3)(x+5)} \)

We can cancel the common factor \( (x + 3) \) from the numerator and denominator:

\[
\frac{(x-1)(x+3)}{(x+3)(x+5)} = \frac{x-1}{x+5}, \quad \text{where} \, x \neq -3 \, (\text{to avoid division by zero})
\]

Thus, the simplified form is:

\[
\frac{x-1}{x+5}
\]

---

### Summary of Simplified Results:

1. \( \frac{7}{x} \)
2. \( \frac{5a^4}{2b} \)
3. \( \frac{2}{x+3} \)
4. \( \frac{x-1}{x+5} \)

Let me know if you need further details or explanations!

Here are some related questions you could consider:
1. How do you simplify expressions with negative exponents?
2. What are the conditions to cancel terms in fractions?
3. How do you simplify a ratio of polynomials?
4. What happens when the numerator and denominator have common factors?
5. How do powers of variables interact in multiplication and division?

**Tip**: Always check for values that make the denominator zero when simplifying fractions involving variables.

Simplify the following expressions:

1) \( \frac{7x^3}{x^4} \)
2) \( \frac{15a^6b}{6a^2b^2} \)
3) \( \frac{2(x-2)}{(x-2)(x+3)} \)
4) \( \frac{(x-1)(x+3)}{(x+3)(x+5)} \)

Learn how to simplify rational expressions involving exponents and common factors step by step. These problems cover algebraic concepts, exponent rules, and factorization, suitable for students in Grades 8-10.

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