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The given mathematical expression is:

\[
\frac{15m^4n^5}{12m^4n^4} \cdot \frac{14m}{3n}
\]

Let's break this down and simplify step by step.

### Step 1: Simplify the first fraction \(\frac{15m^4n^5}{12m^4n^4}\)
- Simplify the coefficients: \(\frac{15}{12} = \frac{5}{4}\)
- For the \(m\) terms: \(\frac{m^4}{m^4} = 1\)
- For the \(n\) terms: \(\frac{n^5}{n^4} = n^{5-4} = n^1 = n\)

Thus, the simplified first fraction becomes:
\[
\frac{5n}{4}
\]

### Step 2: Simplify the second fraction \(\frac{14m}{3n}\)
This fraction cannot be simplified further, so it remains:
\[
\frac{14m}{3n}
\]

### Step 3: Multiply the two simplified fractions
Now, multiply the two results:
\[
\frac{5n}{4} \cdot \frac{14m}{3n}
\]

- Multiply the numerators: \(5n \cdot 14m = 70mn\)
- Multiply the denominators: \(4 \cdot 3n = 12n\)

Thus, we get:
\[
\frac{70mn}{12n}
\]

### Step 4: Simplify the result
- Cancel out the \(n\) terms: \(\frac{mn}{n} = m\)
- Simplify the coefficient: \(\frac{70}{12} = \frac{35}{6}\)

Thus, the final simplified result is:
\[
\frac{35m}{6}
\]

Would you like any more details on this solution? Here are some related questions to consider:

1. What happens if the powers of \(n\) are different in the second fraction?
2. How do you handle exponents in the numerator and denominator?
3. Can this technique be applied to more complex rational expressions?
4. What is the general rule for simplifying fractions with variables?
5. How can you verify your result by substituting values?

**Tip:** Always simplify the coefficients and variables separately to avoid confusion!

Simplify the expression: (15m^4n^5 / 12m^4n^4) * (14m / 3n)

Learn how to simplify the rational expression (15m^4n^5 / 12m^4n^4) * (14m / 3n) using algebraic rules for exponents and fractions. This step-by-step solution is ideal for students in Grades 8-10.

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