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Let's solve the given problem step-by-step.

The expression to simplify is:
\[
\frac{21y^6x^4 - 18y^4x + 24y^6x^5}{3y^5x^2}
\]

### Step 1: Separate Terms and Divide Each by \(3y^5x^2\)
We can rewrite this as three separate fractions:
\[
\frac{21y^6x^4}{3y^5x^2} - \frac{18y^4x}{3y^5x^2} + \frac{24y^6x^5}{3y^5x^2}
\]

### Step 2: Simplify Each Term Individually
1. **First Term**:
   \[
   \frac{21y^6x^4}{3y^5x^2} = \frac{21}{3} \cdot \frac{y^6}{y^5} \cdot \frac{x^4}{x^2} = 7y^{6-5}x^{4-2} = 7yx^2
   \]

2. **Second Term**:
   \[
   \frac{18y^4x}{3y^5x^2} = \frac{18}{3} \cdot \frac{y^4}{y^5} \cdot \frac{x}{x^2} = 6y^{4-5}x^{1-2} = 6y^{-1}x^{-1} = \frac{6}{yx}
   \]

3. **Third Term**:
   \[
   \frac{24y^6x^5}{3y^5x^2} = \frac{24}{3} \cdot \frac{y^6}{y^5} \cdot \frac{x^5}{x^2} = 8y^{6-5}x^{5-2} = 8yx^3
   \]

### Step 3: Combine the Results
Putting it all together, the simplified expression is:
\[
7yx^2 - \frac{6}{yx} + 8yx^3
\]

Would you like a more detailed explanation or have any questions?

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Here are 5 additional questions that extend this topic:

1. How do you simplify expressions involving negative exponents?
2. What is the rule for dividing terms with the same base but different exponents?
3. How can you further simplify expressions with fractions in algebra?
4. How do operations on powers differ between multiplication and division?
5. Can this expression be factored further if needed?

**Tip**: When simplifying complex fractions, try breaking them down into separate fractions to handle each part individually.

Divide \(\frac{21y^6x^4 - 18y^4x + 24y^6x^5}{3y^5x^2}\) and simplify as much as possible.

Learn how to divide and simplify the polynomial expression \(\frac{21y^6x^4 - 18y^4x + 24y^6x^5}{3y^5x^2}\). This solution involves key concepts in algebra, such as polynomial division and exponent rules, making it suitable for Grades 9-11.

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