To simplify the expression \(\sqrt{20 a^{4} b^{6} c^{3}}\), we need to break it down into its prime factors and apply the square root to each part separately. We can simplify the square root of each variable by using the property \(\sqrt{x^2} = x\).
Step 1: Factor the Expression
We start with the expression \( \sqrt{20 a^{4} b^{6} c^{3}} \). We can factor \( 20 \) into its prime components:
\[
20 = 4 \times 5 = 2^2 \times 5
\]
Thus, we can rewrite the expression as:
\[
\sqrt{20 a^{4} b^{6} c^{3}} = \sqrt{2^2 \times 5 \times a^{4} \times b^{6} \times c^{3}}
\]
Step 2: Apply the Square Root
Using the property \( \sqrt{x \cdot y} = \sqrt{x} \cdot \sqrt{y} \), we can separate the square root:
\[
\sqrt{20 a^{4} b^{6} c^{3}} = \sqrt{2^2} \cdot \sqrt{5} \cdot \sqrt{a^{4}} \cdot \sqrt{b^{6}} \cdot \sqrt{c^{3}}
\]
Calculating each square root gives us:
\[
\sqrt{2^2} = 2, \quad \sqrt{a^{4}} = a^{2}, \quad \sqrt{b^{6}} = b^{3}, \quad \sqrt{c^{3}} = c^{\frac{3}{2}} = c^{1} \cdot \sqrt{c}
\]
Step 3: Combine the Results
Now we can combine all the simplified parts:
\[
\sqrt{20 a^{4} b^{6} c^{3}} = 2 \cdot \sqrt{5} \cdot a^{2} \cdot b^{3} \cdot c^{1} \cdot \sqrt{c}
\]
This simplifies to:
\[
2 a^{2} b^{3} c \sqrt{5c}
\]
Final Answer
Thus, the simplified form of the expression is:
\[
\boxed{2 a^{2} b^{3} c \sqrt{5c}}
\]