https://file.aiquickdraw.com/mathbot/file/17268874494662z4t9maj.jpg

The given expression is:

\[
\sqrt{18a^5}
\]

We are asked to simplify this expression, assuming \( a \geq 0 \).

### Step-by-step simplification:

1. **Factor the radicand**: 
   \[
   \sqrt{18a^5} = \sqrt{18} \times \sqrt{a^5}
   \]
   We can simplify \( \sqrt{18} \). Since \( 18 = 9 \times 2 \), we have:
   \[
   \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}
   \]

2. **Simplify \( \sqrt{a^5} \)**:
   We break \( a^5 \) into \( a^4 \times a \), where \( a^4 \) is a perfect square:
   \[
   \sqrt{a^5} = \sqrt{a^4 \times a} = \sqrt{a^4} \times \sqrt{a} = a^2 \times \sqrt{a}
   \]

3. **Combine the results**:
   Now, substituting back into the original expression:
   \[
   \sqrt{18a^5} = 3\sqrt{2} \times a^2 \times \sqrt{a}
   \]
   This simplifies to:
   \[
   3a^2\sqrt{2a}
   \]

Thus, the simplified expression is:
\[
3a^2\sqrt{2a}
\]

Would you like any further details or clarification?

### Related Questions:
1. How do you simplify square roots involving both constants and variables?
2. What is the process for factoring numbers under a square root?
3. How does the assumption \( a \geq 0 \) affect the solution?
4. Can this expression be further simplified if we know specific values for \( a \)?
5. How does the exponent rule apply to square roots?

### Tip:
Always look for perfect squares when simplifying square roots—they can greatly reduce the complexity of the expression.

Simplify. Assume a is greater than or equal to zero. \( \sqrt{18a^5} \)

Learn how to simplify the square root expression √(18a^5) by breaking it down into factors, applying the product rule of radicals, and using exponent rules. Detailed solution with key concepts suitable for students in Grades 9-11.

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