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.