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Let's perform the indicated operations for each expression one by one.

### 1. Expression: \(\frac{r^{7/5}}{r} \div \frac{dr}{r^3}\)

We already solved this one. The simplified form is:
\[
\frac{r^{17/5}}{dr}
\]

### 2. Expression: \(\frac{u^{5/2}x}{y} \div \frac{14x^2}{y^4}\)

To divide fractions, we multiply by the reciprocal of the divisor:

\[
\frac{u^{5/2}x}{y} \div \frac{14x^2}{y^4} = \frac{u^{5/2}x}{y} \times \frac{y^4}{14x^2}
\]

Simplify the expression:

\[
= \frac{u^{5/2}x \cdot y^4}{14x^2 \cdot y}
\]

\[
= \frac{u^{5/2}xy^4}{14x^2y}
\]

Next, simplify the powers of \(x\) and \(y\):

- \(x\) terms: \(x/x^2 = x^{1 - 2} = x^{-1} = \frac{1}{x}\)
- \(y\) terms: \(y^4/y = y^{4-1} = y^3\)

Substitute back into the expression:

\[
= \frac{u^{5/2}y^3}{14x}
\]

So, the simplified form is:

\[
\frac{u^{5/2}y^3}{14x}
\]

### 3. Expression: \(\frac{m}{2} \div \frac{m - 1}{m - 2}\)

Again, dividing by a fraction is the same as multiplying by its reciprocal:

\[
\frac{m}{2} \div \frac{m - 1}{m - 2} = \frac{m}{2} \times \frac{m - 2}{m - 1}
\]

Multiply the numerators and denominators:

\[
= \frac{m(m - 2)}{2(m - 1)}
\]

So, the simplified form is:

\[
\frac{m(m - 2)}{2(m - 1)}
\]

### 4. Expression: \(\frac{k^2 - 25}{k \cdot 15} \div \frac{k - 5}{k^2 \cdot 16k \cdot 5}\)

Start by factoring \(k^2 - 25\):

\[
k^2 - 25 = (k - 5)(k + 5)
\]

Now, write the expression with the factored form:

\[
\frac{(k - 5)(k + 5)}{k \cdot 15} \div \frac{k - 5}{k^2 \cdot 16k \cdot 5}
\]

Change the division to multiplication by the reciprocal:

\[
\frac{(k - 5)(k + 5)}{k \cdot 15} \times \frac{k^2 \cdot 16k \cdot 5}{k - 5}
\]

Simplify by canceling out \(k - 5\) from the numerator and denominator:

\[
\frac{(k + 5)}{k \cdot 15} \times \frac{k^2 \cdot 16k \cdot 5}{1}
\]

Simplify the expression:

\[
= \frac{(k + 5)(16k^3 \cdot 5)}{k \cdot 15}
\]

\[
= \frac{(k + 5)(80k^3)}{15k}
\]

Simplify further:

\[
= \frac{80k^2(k + 5)}{15}
\]

Divide 80 by 15:

\[
= \frac{16k^2(k + 5)}{3}
\]

Thus, the simplified form is:

\[
\frac{16k^2(k + 5)}{3}
\]

Would you like further details or have any questions?

Here are some related questions you might find interesting:
1. How do you factor polynomials like \(k^2 - 25\)?
2. What are the rules for multiplying fractions in algebra?
3. How do you handle expressions with multiple variables?
4. Can you explain how to simplify complex fractions step by step?
5. What is the process for dividing algebraic fractions?

**Tip:** When simplifying expressions with fractions, always look for common factors in the numerator and denominator that can be canceled out to simplify the expression.

Learn how to simplify algebraic expressions involving fraction division step-by-step. This problem covers key concepts in algebraic simplification and fraction manipulation, suitable for high school students.

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