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To simplify the given expression \(\sqrt{18 \cdot a^4 \cdot c^9}\), follow these steps:

1. **Combine under one radical:**
   \[
   \sqrt{18 \cdot a^4 \cdot c^9}
   \]

2. **Express the product inside the radical:**
   \[
   \sqrt{18} \cdot \sqrt{a^4} \cdot \sqrt{c^9}
   \]

3. **Simplify each term separately:**
   - \(\sqrt{18}\): Factorize 18 to \(18 = 9 \cdot 2\), so \(\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}\).
   - \(\sqrt{a^4}\): Since \(a^4\) is a perfect square, \(\sqrt{a^4} = a^{4/2} = a^2\).
   - \(\sqrt{c^9}\): Rewrite \(c^9\) as \(c^8 \cdot c\), then \(\sqrt{c^9} = \sqrt{c^8 \cdot c} = \sqrt{c^8} \cdot \sqrt{c} = c^4 \cdot \sqrt{c}\).

4. **Combine the simplified terms:**
   \[
   3\sqrt{2} \cdot a^2 \cdot c^4 \cdot \sqrt{c}
   \]

5. **Combine the radicals:**
   \[
   3a^2c^4 \sqrt{2c}
   \]

So, the simplified form of \(\sqrt{18 \cdot a^4 \cdot c^9}\) is:
   \[
   3a^2c^4\sqrt{2c}
   \]

Would you like further details or have any questions?

Here are some related questions to further your understanding:

1. How would you simplify \(\sqrt{50 \cdot b^6}\)?
2. Can you explain the properties of square roots used in the simplification process?
3. What is the simplified form of \(\sqrt{72 \cdot x^{10}}\)?
4. How would you handle \(\sqrt{18 \cdot y^9}\) if \(y\) is a variable?
5. How can you simplify \(\sqrt{3a^6 \cdot b^4}\)?
6. How do you simplify \(\sqrt{4x^8 \cdot 9y^6}\)?
7. What steps would you follow to simplify \(\sqrt{8c^5 \cdot 2d^7}\)?
8. Can you describe the process of simplifying \(\sqrt{27 \cdot z^12}\)?

**Tip:** When simplifying square roots, always look for perfect squares inside the radical to simplify your expression more efficiently.

Learn how to simplify the expression √(18 * a^4 * c^9) step-by-step. This problem involves combining terms under one radical and simplifying as much as possible outside the square root. Suitable for advanced high school students and beyond.

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