\frac{\left(5x^{3}y^{3}\right)^{2}}{\left(x^{2}y\right)^{3}}

To simplify the expression \(\frac{\left(5x^{3}y^{3}\right)^{2}}{\left(x^{2}y\right)^{3}}\), follow these steps:

1. **Expand the numerator**: 
   \[
   \left(5x^3y^3\right)^2 = 5^2 \cdot \left(x^3\right)^2 \cdot \left(y^3\right)^2 = 25x^{6}y^{6}
   \]

2. **Expand the denominator**:
   \[
   \left(x^2y\right)^3 = \left(x^2\right)^3 \cdot y^3 = x^{6}y^{3}
   \]

3. **Form the new expression**:
   \[
   \frac{25x^{6}y^{6}}{x^{6}y^{3}}
   \]

4. **Simplify the powers of \(x\)**: Since \(x^6\) is in both the numerator and the denominator, they cancel out:
   \[
   \frac{25x^{6}y^{6}}{x^{6}y^{3}} = 25y^{6-3} = 25y^3
   \]

The simplified expression is \(25y^3\).

Would you like more details, or do you have any questions?

Here are 5 related questions you might find interesting:
1. How would you simplify \(\frac{\left(3x^{4}y^{2}\right)^{2}}{\left(xy^{3}\right)^{3}}\)?
2. What is the result of simplifying \(\frac{\left(2x^{5}y^{2}\right)^{3}}{\left(x^{2}y\right)^{4}}\)?
3. Can you explain how to apply the power rule for exponents when both a base and an exponent are raised to another exponent?
4. How would you simplify \(\frac{\left(7x^{2}y^{4}\right)^{2}}{\left(x^{3}y^{2}\right)^{2}}\)?
5. If \(25y^3 = 100\), what is the value of \(y\)?

**Tip:** When simplifying expressions with multiple variables raised to powers, always expand the powers first before canceling terms in the numerator and denominator.

Learn how to simplify the expression (5x^3y^3)^2 / (x^2y)^3 with a detailed step-by-step solution. This problem involves exponents and the power rule, crucial concepts in algebra and advanced mathematics. Suitable for students in Grades 9-12.

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