Step 1: Simplify the Bases
First, express both numbers with the same base to make comparison easier.

1. Quantity A: $\(200^{400}\)$
- $\(200=2 \times 100\)$, but this doesn't simplify the exponentiation directly.

2. Quantity B: $\(400^{200}\)$
- $\(400=2 \times 200\)$, but again, this doesn't directly help.

Step 2: Rewrite Using Exponent Rules
Instead, express both quantities in terms of powers of 2 and powers of 100:

1. Quantity A: $\(200^{400}=(2 \times 100)^{400}=2^{400} \times 100^{400}\)$
- $\(100=10^2\)$, so $\(100^{400}=\left(10^2\right)^{400}=10^{800}\)$
- Thus, $\(200^{400}=2^{400} \times 10^{800}\)$

2. Quantity B: $\(400^{200}=(4 \times 100)^{200}=4^{200} \times 100^{200}\)$
- $\(4=2^2\)$, so $\(4^{200}=\left(2^2\right)^{200}=2^{400}\)$
- $\(100^{200}=\left(10^2\right)^{200}=10^{400}\)$
- Thus, $\(400^{200}=2^{400} \times 10^{400}\)$

Step 3: Compare the Simplified Forms
Now, compare the two simplified expressions:
- Quantity A: $\(2^{400} \times 10^{800}\)$
- Quantity B: $\(2^{400} \times 10^{400}\)$

Since $2^{400}$ is common to both, we can focus on the remaining parts:
- Quantity A: $\(10^{800}\)$
- Quantity B: $\(10^{400}\)$

Clearly, $\(10^{800}>10^{400}\)$, so:

$$
\(200^{400}>400^{200}\)
$$