consider this problem what is the square root of 1156 given that you know the answer is an integer or a whole number how would you find the answer well in this video we're going to talk about how to do just that but first we need to talk about the patterns of perfect squares so i'm going to write down a few as i write these down notice what you see one squared is one two squared is four three squared is nine four squared is sixteen five squared is twenty twenty-five six squared is equal to thirty-six seven squared is forty-nine and then eight squared is sixty-four nine squared is eighty-one ten squared is a hundred now i'm going to continue eleven squared is one twenty-one next we have 144 and then 169 14 squared is 196. do you see a pattern emerging at this point look at the units digits 16 squared is 256 17 squared is 289 next we have 19 squared which is 361 and finally 20 squared is 400 so notice that these two numbers are the same so anytime you square a number that ends in a a 1 or a 9 the perfect square will have a one as the unit digit you can see that here as well next we have four anytime you square a number with a two or eight the perfect square will end in a four next is nine anytime you have two numbers that end or if you're taking the square of three or seven you're gonna get a number that ends with a nine so understanding this pattern can help us to find the square root of a very large number now anytime we square a number that ends in a 4 or 6 the perfect square will have a unit value a unit digits value of six so 14 squared and 16 squared they're perfect squares both end up in a six and finally if you square a number containing five it will end with a five and if you square a number containing a ten it's going to end with a zero so those are some things that you want to keep in mind so hopefully you wrote that down but now let's apply it to this the square root of 1156. so the first thing we want to do is look at the last digi the six we know that two numbers that will give us a last digit of a six is four and six four squared is sixteen six squared is thirty six so the number has to end in a four or a six now look at the the first two numbers the square root of eleven the square root of eleven is between the square root of nine and the square root of sixteen the square root of 9 is 3 the square root of 16 is 4. 11 is between 9 and 16 so the answer has to be between 3 and 4. so we're going to pick the lower of the two 3. so our final answer is either thirty four or thirty six the question is which one thirty squared three squared is nine so thirty squared is nine hundred forty squared four squared is sixteen so 40 squared is 1600 thus the answer is going to be closer to 900 1156 is closer to 900 than 1600 so we're going to pick the smaller of the two values which is thirty four this is the answer thirty four squared is equal to eleven fifty six if you multiply thirty four by thirty four you'll get eleven fifty six but now let's try another example what is the square root of 2304 so feel free to pause the video and try this problem so what numbers will end in a well we know that 2 squared is 4 and 8 squared is 64. so the fact that we're ending with a 4 means that the last digit is either a 2 or 8. now what about the first digit so we have the square root of 2304 which is between the square root of 1600 and the square root of twenty five hundred what is the square root of sixteen hundred that's sixteen times a hundred the square root of sixteen is four the square root of a hundred is ten so that's forty likewise the square root of 2500 is 50. so the answer is between 40 and 50. well 2304 is less than 2500 so it has to be less than 50 which means we can't use the five we have to use a four as the first digit so the answer is either 42 or 48 now let's pause for a second and think about something 2304 this number is it closer to 1600 or is it closer to 2500 well 2304 is closer to 2500 now the answer should it be closer to 40 or should it be closer to 50. the final answer should be closer to 50. so which one's closer to 50 42 or 48 we know that it's 48 and this is the final answer the square root of 2304 is indeed 48. if you have a calculator you could type it in to confirm that so let's try another one let's try the square root of four thousand four hundred and eighty nine so what perfect squares end with a nine we know that three squared is nine and seven squared is forty nine so the first digit has to be a three or seven now what about the last digit well let's look at the whole number 44.89 6 squared is 36 7 squared is 49. so 44.89 is going to be between the square root of 3 600 and the square root of 4 900. the square root of 3600 we know that the square root of 36 is 6 so the square root of 3600 is the square root of forty nine hundred will be seventy so the answer is between sixty and seventy so we're going to pick the smaller of these two numbers so the first digit is gonna be a six our answer is either 63 or 67 the question is which one is the right answer would you say it's 63 or 67 now 4489 is that closer to 3600 or 4900 well we can clearly see that 44.89 is a lot closer to 4900 so we need to pick the answer that's closer to 70. and that is going to be 67. this is the right answer now let's try a harder problem let's try taking the square root of a five digit number let's say 12996 go ahead and pause the video and work on that so what perfect squares end with a six we know that four squared is 16 and 6 squared is 36 so the last digit will end either in the 4 or in the 6. now we know that the square root of ten thousand is a hundred the square root of a hundred twenty one thousand i mean twelve thousand one hundred or 12100 is 110. if you type in 110 squared you're going to get 1 12 100 and 12 squared is 144 so the square root of 14 400 is going to be 120 and you could check that all you got to do is just add a zero now 12 996 is between these two numbers so we're going to pick the smaller of 11 and 12 the smaller the two numbers so we're going to go with 11. so the answer is either 114 or 116. which one is it going to be now 12 996 is closer to 12 100 than it is to 14 400 so it's going to be closer to 110 than 120 and 114 is closer to 110. so 114 is the answer if you type in 114 squared you're going to get 12 996. now let's work on one more example twenty four thousand six hundred and forty nine go ahead and find the square root of that number so we know three squared is nine and seven squared is 49 so since the last digit is 9 we know the last digit of our answer will end in a 3 or a 7. now let's focus on this number 246. we know that 14 or rather 15 squared is 225 and 16 squared is 256. so the first two digits will be between 15 and 16 but we're going to pick the lower number so that's going to be 15. so the answer is either 153 or 157 so that means that the square root of 22 500 is 150 because the square root of 225 is 15. the square root of 25 600 is going to be 160. now this number twenty four thousand six forty nine is closer to twenty five thousand six hundred than twenty two thousand five hundred so the answer is going to be one hundred and fifty seven and let's confirm that and that is indeed the answer so now you know how to take the square root of a perfect square even if it's a very large perfect square