It's useful in many questions to know the small powers of the smallest primes. If you know that 5^4 = 25^2 = 625, you'll know that the positive square root of 576 is just less than 25, which leads you to the correct answer- 24- quite quickly.

dextar wrote: we know that square root of 200 would lie between 14(196) and 15(225).
Now 200=14 + (200-196)/(225-196)
=14+6/29

dextar, that is not the *exact* square root of 200, it's only an estimate. The square root of any positive integer is either an integer or an *irrational number*, i.e. a number that cannot be written as a fraction involving two integers. root(200) = 10*(root(2)); put that in your calculator and you'll see it's different from 14+6/29.

I like Jason's method above, but it will only be quick when the square root in question has a small prime factor (2, 3, or 5). If there is no small prime factor, I'd estimate, then use the last digit:

root(400) < root(529) < root(625)
20 < root(529) < 25

Note now that 21^2 would end in 1, 22^2 would end in 4, 23^2 would end in 9, and 24^2 would end in 6. The only candidate among whole numbers is 23. Of course, there's no guarantee that the root is a whole number at all, but if you know you're looking for a whole number, this gets you there quickly.

I own the GMAT wrote:how do you calculate the square root of a HUGE number, like 48400?
i get to 20,/13, and then i'm stuck. help?

If possible, divide it up into smaller numbers, the roots of which can be calculated easily. According to the square root rules:

root(48400) = root(484) * root(100) = root(4) * root(121) * root(100)
= 2*11*10
= 220

fstwrx wrote: I was trying to find out a way to take squares of smaller digits inside of the larger number. Adding a 0 to the last digit and multiplying by 2 seems weird but this method works for every 2 digit number. Does anybody know how this method can be used for numbers greater than 2 digits? It seems like it could be useful for finding square roots and squares.

You're actually just using this well-known factoring pattern (which shows up in all kinds of places on the GMAT) -

(x + y)^2 = x^2 + 2xy + y^2

So, using this with x = 10, y = 9, we have:

19^2 = (10 + 9)^2 = 10^2 + 2*10*9 + 9^2 = 10^2 + 9^2 + 2*90

Of course you can use that with any number at all; it's just that the calculations get messier the more complicated the number. Still, something like 101^2 is easy enough to compute:

101^2 = (100 + 1)^2 = 100^2 + 2*100*1 + 1^2 = 10,201

It's still going to be awkward to calculate something like 387^2, but you don't ever need to do that type of thing on the GMAT anyway, fortunately (it would be quite a boring test if you did!).