To simplify the expression \(\sqrt{75x^6y^5}\), follow these steps:

### Step 1: Break down the square root into factors.

\[
\sqrt{75x^6y^5} = \sqrt{75} \cdot \sqrt{x^6} \cdot \sqrt{y^5}
\]

### Step 2: Simplify each square root separately.

- **Simplify \(\sqrt{75}\)**:
  
  \[
  \sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}
  \]

- **Simplify \(\sqrt{x^6}\)**:
  
  \[
  \sqrt{x^6} = x^{\frac{6}{2}} = x^3
  \]

- **Simplify \(\sqrt{y^5}\)**:
  
  Break down \(y^5\) as \(y^4 \cdot y\):
  
  \[
  \sqrt{y^5} = \sqrt{y^4 \cdot y} = \sqrt{y^4} \cdot \sqrt{y} = y^2\sqrt{y}
  \]

### Step 3: Combine the simplified parts.

Putting it all together:

\[
\sqrt{75x^6y^5} = 5x^3y^2\sqrt{3y}
\]

### Final Expression:
\[
\sqrt{75x^6y^5} = 5x^3y^2\sqrt{3y}
\]

Would you like to go over any part of the process or have any further questions?

### Related Questions:
1. How do you simplify the square root of an expression like \(\sqrt{50x^4y^6}\)?
2. What are the steps to rationalize a square root expression?
3. How do you simplify cube roots of algebraic expressions?
4. What are the properties of square roots in algebra?
5. How can you simplify \(\sqrt{48a^7b^3}\)?

### Tip:
When simplifying square roots with variables, factor out perfect squares for easy simplification.

Learn how to simplify the square root of 75x^6y^5 with detailed steps. This problem involves square roots and exponents, focusing on algebraic simplification techniques. Suitable for Grades 9-12.

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