To simplify the expression \(\frac{\sqrt{45xy^2}}{\sqrt{35x^4y}}\), follow these steps:

1. **Simplify the square roots separately**:

   - \(\sqrt{45xy^2} = \sqrt{45} \times \sqrt{x} \times \sqrt{y^2}\)
   - \(\sqrt{35x^4y} = \sqrt{35} \times \sqrt{x^4} \times \sqrt{y}\)

2. **Simplify the individual square roots**:

   - \(\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}\)
   - \(\sqrt{35} = \sqrt{5 \times 7} = \sqrt{5} \times \sqrt{7}\)
   - \(\sqrt{x}\) remains as \(\sqrt{x}\)
   - \(\sqrt{y^2} = y\)
   - \(\sqrt{x^4} = x^2\)
   - \(\sqrt{y}\) remains as \(\sqrt{y}\)

   So:
   \[
   \sqrt{45xy^2} = 3\sqrt{5} \times \sqrt{x} \times y = 3y\sqrt{5x}
   \]
   \[
   \sqrt{35x^4y} = \sqrt{5} \times \sqrt{7} \times x^2 \times \sqrt{y} = x^2\sqrt{35y}
   \]

3. **Divide the simplified expressions**:
   \[
   \frac{3y\sqrt{5x}}{x^2\sqrt{35y}}
   \]

   Simplifying further:
   - Divide the coefficients: \(\frac{3y}{x^2}\)
   - Divide the square roots: \(\frac{\sqrt{5x}}{\sqrt{35y}} = \frac{\sqrt{5x}}{\sqrt{5 \times 7y}} = \frac{\sqrt{5x}}{\sqrt{5}\sqrt{7y}} = \frac{\sqrt{x}}{\sqrt{7y}}\)

   So the final simplified expression is:
   \[
   \frac{3y}{x^2} \times \frac{\sqrt{x}}{\sqrt{7y}} = \frac{3\sqrt{x}}{x^2

Learn how to simplify the division of square roots for the expression √45xy^2 divided by √35x^4y. This detailed step-by-step solution covers concepts such as square root simplification and division, suitable for high school students.

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