If you see something like this on test day, I would probably get as close as possible to an answer via rounding and then look at the answers to see which is closest. That said...here's what I would have done.

When you get to SquareRoot(22.4x10^5)...it is the same as SquareRoot(224x10^4). Pull out the 10^4 as 10^2. The SquareRoot(224=225) = 15. The answer you get is 15 *10^2, which is really close to 1.4 *10^3.

Very similar, but I would have done a different way. Try them and see which one is more intuitive to you, OP. The important key here is that you want to make 10^5 to be "even" like 10^4 or 10^6, so that after square root, you get 10^2 or 10^3 (just divide the powers by 2 since square root is the same as ^0.5)

Square root(22.4x10^5) -> decide which: square root (2.24x10^6) or (224x10^4) -> if you don't know the square root of 225 like the poster up, try the other way, like square root of 2 is about 1.4 something -> so square root (2 something x 10^6) = 1.4 something x 10^3.

I hope it helps!