Article: Once you've memorized your perfect squares, finding the square roots of imperfect squares becomes much easier. Since you already know a dozen or so perfect squares, any number that falls between two of these perfect squares can be found by "whittling away" at an estimate between these values. To start, find the two perfect squares your number is between. Next, determine which of these two numbers it is the closest to. For example, let's say we need to find the square root of 40. Since we've memorized our perfect squares, we can say that 40 is in between 62 and 72, or 36 and 49. Since 40 is greater than 62, its square root will be greater than 6, and since it is less than 72, its square root will be less than 7. 40 is a little closer to 36 than it is to 49, so the answer will probably be a little closer to 6. In the next few steps, we'll narrow our answer down. Once you've picked out two perfect squares that your number is between, it's simply a matter of whittling away at your estimate until you reach an answer you're satisfied with the farther you go, the more accurate your answer is. To start, pick a "tenth place" decimal point for your answer it doesn't have to be correct, but you'll save time if you use common sense to pick one that's close to the right answer. In our example problem, a reasonable estimate for the square root of 40 might be 6.4, since we know from above that the answer is probably a little closer to 6 than it is to 7. Next, square your estimate. Unless you're lucky, you probably won't get your original number you'll either be a little higher than it or a little lower. If your answer is too high, try again with a slightly smaller estimate (and vice versa if it is too low). Multiply 6.4 by itself to get 6.4 6.4 = 40.96, which is slightly higher than original number. Next, since we over-shot our answer, we'll multiply the number one tenth less than our estimate above by itself and to get 6.3 6.3 = 39.69. This is slightly lower than our original number. This means that the square root of 40 is somewhere between 6.3 and 6.4. Additionally, since 39.69 is closer to 40 than 40.96, you know the square root will be closer to 6.3 than 6.4. At this point, if you're happy with your answers, you may want to simply use one of your first guesses as an estimate. However, if you'd like a more accurate answer, all you need to do is pick an estimate for your "hundredths place" that puts this estimate between your first two. Continuing with this pattern, you can get three decimal places for your answer, four, and so on it just depends how far you want to go. In our example, let's pick 6.33 for our two-decimal point estimate. Multiply 6.33 by itself to get 6.33 6.33 = 40.0689. Since this is slightly above our original number, we'll try a slightly lower number, like 6.32. 6.32 6.32 = 39.9424. This is slightly below our original number, so we know that the exact square root is between 6.33 and 6.32. If we wanted to continue, we would keep using this same approach to get an answer that's continually more and more accurate.

What is a summary?
Find non-perfect squares by estimating. Estimate the square root to one decimal point. Multiply your estimate by itself. Continue estimating as needed.