To simplify the expression $ \sqrt{75x^9} $, let's break it down step by step:

### Step 1: Simplify the square root of the constant
We begin with $ \sqrt{75} $.

- $ 75 = 25 \times 3 $, and since $ \sqrt{25} = 5 $, we can simplify:
  \[
  \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}
  \]

### Step 2: Simplify the square root of the variable
Next, we look at $ \sqrt{x^9} $.

- We can rewrite $ x^9 $ as $ x^8 \times x $.
- The square root of $ x^8 $ is $ x^4 $ because $ \sqrt{x^8} = x^{8/2} = x^4 $.

Thus, the square root of $ x^9 $ is:
\[
\sqrt{x^9} = x^4\sqrt{x}
\]

### Step 3: Combine everything
Now, we put the two parts together:
\[
\sqrt{75x^9} = 5x^4\sqrt{3x}
\]

### Final Simplified Form:
\[
\boxed{5x^4\sqrt{3x}}
\]

Would you like further details on any of the steps? 

Here are 5 questions that expand on this topic:
1. How do you simplify higher-order radicals?
2. Can you apply this method to cube roots or higher powers?
3. What happens when variables have negative exponents under a square root?
4. How do you simplify expressions with multiple variables under the square root?
5. What is the general rule for simplifying roots of powers of variables?

**Tip:** Always break down square roots into simpler parts, separating constants and variables for easier simplification.

Simplify the expression: $$ \sqrt{75x^9} $$

Learn how to simplify the radical expression √(75x^9) using properties of radicals and exponents. This step-by-step solution breaks down the process, including simplifying constants and variable powers. Suitable for students in Grades 9-10, this problem covers core algebra concepts.

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