Simplifying Surds

12*sqrt(2) = sqrt(144 * 2) = sqrt(288).

sqrt(363) = sqrt(121 * 3) = 11*sqrt(3).

To simplify the square root, find the largest perfect square factor of 128. Since 128 = 64 * 2, the square root is sqrt(64) * sqrt(2), which equals 8 * sqrt(2).

To simplify sqrt(600), identify the largest perfect square factor, which is 100. Thus, sqrt(600) = sqrt(100 * 6) = 10 * sqrt(6).

The number 147 can be factored into 49 * 3. Since 49 is a perfect square, sqrt(147) = sqrt(49) * sqrt(3) = 7 * sqrt(3).

The square root of 8100 is the square root of (81 * 100). This simplifies to sqrt(81) * sqrt(100) = 9 * 10 = 90.

The factors of 150 are 25 * 6. Therefore, sqrt(150) = sqrt(25) * sqrt(6) = 5 * sqrt(6). Note that options A and C are identical.

The number 63 can be factored into 9 * 7. Thus, sqrt(63) = sqrt(9) * sqrt(7) = 3 * sqrt(7).

The number 80 can be factored into 16 * 5. Thus, sqrt(80) = sqrt(16) * sqrt(5) = 4 * sqrt(5).

The number 50 can be factored into 25 * 2. Thus, sqrt(50) = sqrt(25) * sqrt(2) = 5 * sqrt(2).