How To Find The Square Root of Large Numbers Using The Division Method

in this video we're going to talk about how to find the square root of large numbers using the division method so let's begin by rewriting the problem now the first thing we're going to do is we're going to break up the four numbers into pairs of two now let's think of 22 22 is between the perfect squares 16 and 25 we know that 4 squared is 16 5 squared is 25 but we're going to use the perfect square that is less than 22 that is 16. and so 4 goes into 22 four times 4 times 4 is 16 and then we're going to subtract 22 by 16 which gives us 6. next we're going to bring down the 0 9. now because we have 4 here we're going to add the same number to it so 4 plus 4 is 8 and we're going to leave space here we need to find number that goes here and whatever number that goes there will go here as well but now here's how we could find that number 80 something times something is 609 what is that something so 80 something times something should equal six or nine how can we find that 609 is close to 640 and 64 is perfect square we know that 8 squared is 64. so if this is 80 something 80 times 8 is 640. so we need to try something that's less than eight let's try seven so 87 times seven what is that let's do little math 7 times 7 is 49 carry the 4 8 times 7 is 56 plus 4 that's 60. so 87 times 7 gives us so now we have zero remaining so whatever number we have in this box that's going to go here the final answer is whatever we see here so the square root of 2209 is 47 and that's how you can find it using the division method now let's work on another example for the sake of practice go ahead and find the square root of five thousand one hundred and eighty-four feel free to take minute and use the same method that we used in the last problem so let's begin by rewriting the problem now let's break up the four numbers into pairs of two and let's focus on 51. so what is the perfect square that is less than 51 we know that 6 squared is 36 7 squared is 49 and 8 squared is 64. so the highest perfect square just under 51 is 49 so we're going to use 7 7 times 7 is 49 subtracting 51 by 49 gives us 2 and then we're going to bring down the 84. now we're going to add whatever number we see here to it so 7 plus 7 is 14 and then let's put box here 140 something times something is equal to 284 what is that something well 284 divided by 140 something is about 2 so 2 is the best number to go with so 142 times 2 is clearly 284 so the number that goes here is going to be 2 so the square root of 5184 is 72 and that is the answer let's try another similar example so that you can master this topic go ahead and find the square root of seven thousand two hundred and twenty five feel free to take minute and work on that so let's begin by rewriting the problem and by breaking the four numbers into pairs of two now what perfect square is just under seventy two we know that eight squared is sixty four and nine squared is eighty one eighty 81 exceeds 72 so we're going to use 64. so 8 times 8 is 64. subtracting 72 by 64 gives us 8 and then we're going to bring down the so we have 825 now let's add 8 to 8 which gives us 16 and let's put box so 160 something times something is 8.25 what is that something well the only way we're going to get 5 at the end is if we have 5 to begin with because 5 times 5 is 25 so clearly this must be 5 and we could check it if we multiply 165 by five we get five times five which is twenty-five carry over the two five times six is thirty plus two that's 32 carry over the three five times one is five plus three we get eight five plus three is eight so this gives us the eight twenty five that we need giving us remainder of zero and since we have five in the box that five is going to go here thus the square root of seven thousand two hundred twenty five is eighty five and to check your work you could just multiply eighty five by eighty five and you'll get seventy two twenty five so that's how you could use the long division method to find the square root of perfect square now let's find the square root of five digit number twenty one thousand three hundred and sixteen so let's work on that example so the first thing we're going to do is we're going to write pairs of twos we're going to highlight pairs of twos starting from the left so we're going to focus on the square root of 2. we know that 1 squared is 1 and 2 squared is 4. so perfect square that is less than 2 is 1. so one times one will give us one and then we're going to subtract two minus one is one and then bring down the thirteen the next pair of numbers so we have one thirteen on the left we're going to add one to one giving us two and then we're going to draw box so 20 something times something is equal to or just under 113 what is that something it doesn't have to be exactly equal to 113 because we haven't used 16 yet but we want number that will give us number that is close to 113 but just under it well we can try five 25 times five is one twenty five so that exceeds one thirteen so the next best option is four twenty four times four four times four is sixteen carry over the one two times four is eight plus one that's ninety six so we're gonna go with that 24 times four is ninety six and now we need to subtract so what is 113 minus well we can't do 3 minus 6 because it's going to give us negative number so we're going to borrow 1. this is going to change to 0 and that becomes 13. 13 minus 6 is 7 10 minus 9 is 1. so we get 17. now 24 and 4 we're going to add them to get 28 and then we're going to create new box now we also need to bring down the 16 so we get 17 16. 28 or 280 something times something is equal to 17 16. what is that number well the fact that this ends in six tells us that it might be six also if we round up 28 to 30 and 171 to 180 we know that 30 times 6 is 180 so 6 is good number to start with so let's try 286 times 6 to see if it's going to give us our desired number 6 times 6 is 36 carry over the 3 8 times 6 is 48 plus 3 that's 51 carry over the 5 2 times six is twelve plus five that's seventeen sixteen so this is going to work that's the nuts the next time we need to put up top is the six forgot to put the four so it should be 146 because four is in the first box and then six is in the second box the square root of is 146. so that's the answer for this example now let's work on one more example let's find the square root of three hundred twenty nine thousand four hundred and seventy go ahead and work on that example so let's break up the six numbers into pairs of two so first let's focus on the square root of 32 we know that 4 squared is 16 5 squared is 25 6 squared is 36 so 25 is the highest perfect square under we're going to go with five five times five is 25 32 minus 25 is 7 and then we're going to bring down the 94. on the left we're going to add 5 to 5. that's going to give us 10 and then we're going to put box so 100 and something times something is close to but less than seven ninety four well it can't be eight 108 times eight is going to be more than eight hundred so the best number to pick is seven 107 times 7 is going to be 7 times 7 is 49 carry the 4. 7 times 0 is 0 plus 4 is 4 and then 7 times 1 is 7. so 107 times 7 will give us seven hundred and forty nine next we need to subtract so what is seven ninety four minus seven forty nine we need to borrow one so the nine becomes eight the four becomes fourteen fourteen minus nine is five eight minus four is four the sevens will cancel so this will give us forty five on the left we need to add 107 plus 7 is 114 and let me not forget to put the 7 here like did last time now we need to add another box and also we need to bring down the 76 so 1140 something times something is equal to 45.76 what is that something well we know that one thousand times four will give us four thousand so four is the best number to go with eleven forty four times four and also the fact that this ends with six means that this could be six or four because four squared ends in six and six squared ends in six but one thousand times six is going to give us six thousand something so we don't want to use six in this example four is the best option so 4 times 4 is 16 carry over the 1 4 times 4 is 16 again plus 1 that's going to be 17 carry over another one 1 times 4 is 4 plus 1 is 5. bring down the last mean don't do that four times one is four so we get forty five seventy six eleven forty four times four is this number and so we have remainder of zero thus we can take our last number and put it here so the square root of 329 476 is 574. that is the answer so now you know how to find the square root of large number using the division method