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.