Indian Math Online recently received a request for help on this math question:
How do I manually determine the square root of 2795?
There are a number of ways of solving this problem….without using a calculator! First you have to understand what exactly a “square root of a number” is. A square root of a number is a special number that when multiplied by its self (or squared as we say), the result is the original number. Take the number 9 for example. The number 3 multiplied by itself gives a result of 9 (3 x 3 = 9). Another way of saying this is 3 squared is 9 (32 = 9). Therefore, 3 is the square root of 9 (3 = √9).
On a multiple choice test where 4 possible answers are given, you could easily just square the 4 answers to see which one is the correct answer. However, that is not very challenging and it takes away all the fun. So if you are not given 4 possible answers to choose, and you don’t have a calculator, it is time to learn how to manually calculate square roots.
GUESSING
The first real method would be to make educated guesses! Most students learn what are called “perfect squares” when they are in middle school. Perfect squares are integers that have integers as a square root. The number 9 is a good example of this because its square root is 3. The number 10 is not a perfect square because its square root is 3.1623. Students memorize these perfect squares, usually as high as 625 which has a square root of 25. So to apply the guessing method to finding the square root of 155, knowing that the two closest perfect squares are 144 (which has square root of 12) and 169 (which has a square root of 13), it is safe to say that the square root of 155 is somewhere between 12 and 13. That is as far as guessing can take you.
PRIME FACTORIZATION
The prime factorization method only works when trying to find the square root of a number which happens to be a perfect square. So unless the question specifies “Find the square root of the perfect square 784”, this method may be a waste of your time trying to use it properly. To find the square root of 784 which is in fact a perfect square, you need to break it down into its smallest factors. The number 784 can be factored into 2 x 2 x 2 x 2 x 7 x 7 (because 2 x 2 x 2 x 2 x 7 x 7 = 784). The next step would be to group these smallest factors into pairs: 2 x 2 is a pair, 2 x 2 is a pair, and 7 x 7 is a pair. If you multiply one of each pair together, you get 2 x 2 x 7 = 28. The square root of 784 is 28.
LONG DIVISION
Finding square roots by long division can be the most difficult method to understand at first, but it will always work for any number, even decimals! To solve the question asked by our student, “what is the square root of 2795”, we must first group 2795 into pairs of 2 digits (27 and 95) and write it like a long division problem:
______
| 27 95 NOTE: If the number were 12549, it would be grouped as 1, 25, and 49
STEP 1: Find the largest number whose square is equal to or less than the first pair. In this case, that number would be 5 because its square is 25 which is less than 27. The number 6 is too big because its square is 36 which is greater than 27, so 5 is the largest number. We now write the 5 in three spots, and its square of 25 in one spot:
5
5 |27 95
5 25
STEP 2: Then on the left side you add the numbers, and on the right side you subtract them:
5
5 | 27 95
+ 5 -25
10 2
STEP 3: Next you want to carry down the next pair of numbers, 95, and put an x after the number 10.
5
5 | 27 95
+ 5 -25
10x 295 NOTE: 10x represents a 3-digit number, and the remainder here is 295
STEP 4: You now want to determine the largest number x such that the number 10x times x is equal to or less than the remainder. In this case, that largest number x is 2 because 102 times 2 = 204 which is equal to or less than 295. The number 3 gives 103 times 3 = 309 which is greater than 295, so 2 is correct. Now write the 2 in three spots, and the 204 from earlier in one spot:
5 2
5 | 27 95
+ 5 -25
102 295
2 204
STEP 5: Then on the left side you add the numbers, and on the right side you subtract them:
5 2
5 | 27 95
+ 5 -25
102 295
+ 2 -204
104 91
STEP 6: You have run out of numbers to bring down, so bring down two zeros (00) which are after the decimal points. Also put an x after the number 104.
5 2 .
5 | 27 95.00
+ 5 -25
102 295
+ 2 -204
104x 9100
At this point you would continue repeating steps 4, 5, and 6 until you have a remainder of zero (0), or you have gotten as many numbers after the decimal as required. The square root of 2795 is 52.86775955…. and keeps going, so usually when you are asked to find the square root manually of a number such as 2795, the question will say to only go as far as 2 or 3 decimal places.
Indian Math Online (IMO) has materials that explain these methods in further detail and with more examples. You can see IMO’s “Solution Explanation” on how to perform all of these methods by going here.
Or you can watch one of IMO’s learning assignments on this topic here.
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