Arithmetic » Simplifying square roots

Simplifying (square) roots is based on the following rule:

square root(ab) = square root(a) × square root(b)


Examples
square root(45) = square root(9×5) = square root(9) × square root(5) = 3 × square root(9) = 3 square root(9)
square root(150) = square root(25×6) = square root(25) × square root(6) = 5 × square root(6) = 5 square root(6)

Note: Take a square as large as possible from the number:

So not: square root(72) = square root(9×8) = square root(9) × square root(8) = 3 × square root(8) = 3 square root(8)
but:square root(72) = square root(36×2) = square root(36) × square root(2) = 6 × square root(2) = 6 square root(2)

What is behind 'So not:' is possible and mathematically correct, but what is wrong is the fact that 3 square root(8) can be simplified further.
The square root(8) can be simplified to square root(4×2).
So: 3 square root(8) = 3 square root(4×2) = 3 × square root(4) × square root(2) = 3 × 2 × square root(2) = 6 square root(2)