Find the largest perfect square in 200, which is 100. Write as sqrt(100 * 2). Since sqrt(100) is 10, the simplified surd is 10 sqrt(2).

The largest perfect square factor of 32 is 16. Express it as sqrt(16 * 2). Extract the square root of 16 to get 4, leaving 4 sqrt(2).

The highest perfect square factor of 12 is 4. Rewrite as sqrt(4 * 3). Separate into sqrt(4) * sqrt(3), which evaluates to 2 sqrt(3).

The largest perfect square factor of 75 is 25. Break it down into sqrt(25 * 3). Taking the square root of 25 leaves 5 outside the root, resulting in 5 sqrt(3).

Identify the largest perfect square dividing 48, which is 16. Rewrite sqrt(48) as sqrt(16 * 3). This simplifies to sqrt(16) * sqrt(3) = 4 sqrt(3).

The largest perfect square factor of 50 is 25. Rewrite as sqrt(25 * 2) = sqrt(25) * sqrt(2). The square root of 25 is 5, giving 5 sqrt(2).

Find the largest perfect square factor of 18, which is 9. Rewrite sqrt(18) as sqrt(9 * 2). Using the multiplication property of surds, split it into sqrt(9) * sqrt(2). This simplifies to 3 sqrt(2).

First, handle the negative exponent by taking the reciprocal: 1 / (16^(1/4)). Then, find the 4th root of 16, which is 2. The expression simplifies to 1/2.

Apply the exponent to both the numerator and the denominator. The cube root of 1 is 1, and the cube root of 8 is 2. Thus, the result is 1/2.

The exponent 1/4 indicates the 4th root. We are looking for a number that multiplied by itself four times equals 10,000. 10 * 10 * 10 * 10 = 10,000, so the answer is 10.