This method uses a number's factors to find a number's square root (depending on the number, this can be an exact numerical answer or a close estimate). A number's factors are any set of other numbers that multiply together to make it. For instance, you could say that the factors of 8 are 2 and 4 because 2 × 4 = 8. Perfect squares, on the other hand, are whole numbers that are the product of other whole numbers. For instance, 25, 36, and 49 are perfect squares because they are 52, 62, and 72, respectively. Perfect square factors are, as you may have guessed, factors that are also perfect squares. To start finding a square root via prime factorization, first, try to reduce your number into its perfect square factors.  Let's use an example. We want to find the square root of 400 by hand. To begin, we would divide the number into perfect square factors. Since 400 is a multiple of 100, we know that it's evenly divisible by 25 - a perfect square. Quick mental division lets us know that 25 goes into 400 16 times. 16, coincidentally, is also a perfect square. Thus, the perfect square factors of 400 are 25 and 16 because 25 × 16 = 400. We would write this as: Sqrt(400) = Sqrt(25 × 16) The product property of square roots states that for any given numbers a and b, Sqrt(a × b) = Sqrt(a) × Sqrt(b). Because of  this property, we can now take the square roots of our perfect square factors and multiply them together to get our answer. In our example, we would take the square roots of 25 and 16. See below:  Sqrt(25 × 16) Sqrt(25) × Sqrt(16) 5 × 4 = 20 In real life, more often than not, the numbers you'll need to find square roots for won't be nice round numbers with obvious perfect square factors like 400. In these cases, it may not be possible to find the exact answer as an integer. Instead, by finding any perfect square factors that you can, you can find the answer in terms of a smaller, simpler, easier-to-manage square root. To do this, reduce your number to a combination of perfect square factors and non-perfect square factors, then simplify. Let's use the square root of 147 as an example. 147 isn't the product of two perfect squares, so we can't get an exact integer value as above. However, it is the product of one perfect square and another number - 49 and 3. We can use this information to write our answer in simplest terms as follows:  Sqrt(147) = Sqrt(49 × 3) = Sqrt(49) × Sqrt(3) = 7 × Sqrt(3) With your square root in simplest terms, it's usually fairly easy to get a rough estimate of a numerical answer by guessing the value of any remaining square roots and multiplying through. One way to guide your estimates is to find the perfect squares on either side of the number in your square root. You'll know that the decimal value of the number in your square root is somewhere between these two numbers, so you'll be able to guess in between them. Let's return to our example. Since 22 = 4 and 12 = 1, we know that Sqrt(3) is between 1 and 2 - probably closer to 2 than to 1. We'll estimate 1.7. 7 × 1.7 = 11.9 If we check our work in a calculator, we can see that we're fairly close to the actual answer of 12.13. This works for larger numbers as well. For example, Sqrt(35) can be estimated to be between 5 and 6 (probably very close to 6). 52 = 25 and 62 = 36. 35 is between 25 and 36, so its square root must be between 5 and 6. Since 35 is just one away from 36, we can say with confidence that its square root is just lower than 6. Checking with a calculator gives us an answer of about 5.92 - we were right. Finding perfect square factors isn't necessary if you can easily determine a number's prime factors (factors that are also prime numbers). Write your number out in terms of its lowest common factors. Then, look for matching pairs of prime numbers among your factors. When you find two prime factors that match, remove both these numbers from the square root and place one of these numbers outside the square root.  As an example, let's find the square root of 45 using this method. We know that 45 = 9 × 5 and we know that 9 = 3 × 3. Thus, we can write our square root in terms of its factors like this: Sqrt(3 × 3 × 5). Simply remove the 3's and put one 3 outside the square root to get your square root in simplest terms: (3)Sqrt(5). From here, it's simple to estimate. As one final example problem, let's try to find the square root of 88:  Sqrt(88) = Sqrt(2 × 44) = Sqrt(2 × 4 × 11) = Sqrt(2 × 2 × 2 × 11). We have several 2's in our square root. Since 2 is a prime number, we can remove a pair and put one outside the square root. = Our square root in simplest terms is (2) Sqrt(2 × 11) or (2) Sqrt(2) Sqrt(11). From here, we can estimate Sqrt(2) and Sqrt(11) and find an approximate answer if we wish.
अपनी संख्या को पूर्ण वर्ग फ़ैक्टर्स में विभाजित करें: अपने परफेक्ट वर्ग फ़ैक्टर्स का वर्गमूल लें: यदि आपकी संख्या के परफेक्ट फ़ैक्टर्स नहीं बन सकते हैं, तो अपने उत्तर को सरलतम टर्म्स तक छोटा करें: यदि आवश्यक हो तो अनुमान लगाएँ: पहले स्टेप के रूप में, अपनी संख्या को छोटा करके लोवेस्ट कामन फैक्टर में बदल लें: