hey this is preshtalwalker in this video i will teach you how to calculate the square root of a perfect square in your head so when would the skill have ever been useful here's the entrance examination for mit in the year 1876 question three is to extract the square root of 64 over 121 and of 11.56 the square root of the fraction is straightforward it will be equal to 8 over 11. but how would you calculate the square root of 11.56 remember in 1876 students did not have pocket calculators that would give an answer instantly instead they would have learned various algorithms to solve this type of problem in this video i will teach you one such method to get started you need to memorize the squares of the numbers from 0 to 9. then focus on the last digit of each of the squares notice that some of these squares end in the same last digit 1 squared and 9 squared both ended one two squared and eight squared both end in four three squared and seven squared both end in nine and four squared and six squared both end in six the square of zero ends in zero that's different from all the other squares listed here and the square of 5 ends in a 5 and that's different from all the other squares listed here so i will illustrate the method with an example let's calculate the square root of sixteen hundred first look at the last digit of sixteen hundred this is zero so you now want the square whose last digit ends in zero there's only one option which is zero squared is equal to zero so we put a zero over here then we cross out the hundreds and units digits we're left with the number sixteen we now want to find the square that's closest to 16 without exceeding 16. in this case 4 squared is equal to 16 exactly so we will write the digit 4 here thus the answer is equal to 40. now this was a simple example you probably could have calculated it without this method but this method will show its value in more difficult examples so let's go ahead with another example let's calculate the square root of 4225 the last digit is 5 and that corresponds to the last digit of 5 squared which is equal to 25 so we write the digit 5 here we then cross out the last two digits we now want the square that's closest to 42 without going over so that will be six squared which is equal to thirty-six so we write a six over here thus the square root of forty-two twenty-five is equal to sixty-five now let's go to a little more complicated example let's calculate the square root of eight hundred and forty-one the last digit is one and that could either correspond to one squared which ends in a one or it could correspond to 9 squared which ends in a 1 as well so we have 2 options of 1 and 9. we'll sort this out in a little bit we'll continue by crossing out the last two digits now we want the square that's closest to 8 without going over so this will be 2 squared which is equal to 4. so we write a 2 here so now we need to go to 2 options we either have 841 is equal to 21 squared or 29 squared to distinguish between the two we'll use a genius method we will calculate the square of 25 which is in between 21 and 29 so there's a trick to calculating the square of a number that ends in 5 you take the part of the number that's not 5 in this case 2 and you multiply it by 1 more than itself so we take 2 multiplied by three which is equal to six then you append a twenty-five so twenty-five squared is equal to six twenty-five notice six twenty-five is smaller than eight hundred forty-one so 25 squared is too little so it must actually be the case that 841 is equal to 29 squared thus we take the larger option of 9. so the square root of 841 is equal to 29. this may sound a little complicated but it'll get easier as we work through more examples let's now calculate the square root of 39.69 we go to the last digit of nine and we know that it's either three or seven we then cross out the last two digits and we look for the square that's closest to 39 without going over this will be 6 squared which is equal to 36 so we now need to differentiate between 63 and 67 we will calculate the square of 65 which is right in the middle so we take 6 and multiply it by one more than itself to get 42. now notice if we have 4225 that'll be larger than 3969 so 65 squared is too large so must instead be equal to 63 squared so thus we know the square root of 3969 is equal to 63. wow let's now calculate the square root of 1156 from the last digit we know the options are either four or six we then cross out the last two digits we look for the square closest to 11 without going over which will be three we now need to differentiate between 34 and 36 so we take 3 multiplied by 1 more than itself which is 12. now 12 is larger than 11 so we need to pick the smaller option of 4. thus the square root of 1156 is equal to 34. we can adapt this solution to calculate the square root of 11.56 this will be equal to 3.4 now let's calculate the square root of sixteen thousand three hundred eighty-four from the last digit we know the two options are either two or eight then we cross out the last two digits we want the square that's closest to 163 without exceeding 163 this will be 12 squared which is 144. we now take 12 multiplied by 1 more than itself which is 156 then we see that 156 is smaller than 163 so we take the larger option of eight thus the square root of sixteen thousand three hundred eighty four is equal to one hundred and twenty eight wow i will mention that you can also adapt this method to calculate the cube root of a number you of course first have to memorize the cubes of the numbers from 0 to 9. then the procedure is very similar in fact it's even easier because each of the last digits is different so we look for the cube whose last digit is equal to 4 the only option is 4 cubed which is equal to 64. so we put down a 4 here we then cross out the last 3 digits we look for the cube that's closest to 39 without going over this will be equal to 3 cubed which is 27. so we put a 3 here thus the cube root of 39 304 is equal to 34. amazing now in the present day we often take arithmetic for granted because we have pocket calculators and powerful computers but historically the new calculating methods were revolutionary let's go back to the early 1200s in europe when roman numerals were the standard practice good luck trying to add multiply or calculate the square root of roman numerals in your head calculations were instead done on an abacus and translated back and forth to roman numerals but with indian numerals calculations could be done directly from the numerals the adoption of the indian numerals didn't happen right away there was actually a competition between the two systems and it took hundreds of years until indian numerals became the standard i will close with a quote from the great french mathematician laplace it is india that gave us the ingenious method of expressing all numbers by the means of 10 symbols each symbol receiving a value of position as well as an absolute value a profound and important idea which appears so simple to us now that we ignore its true merit but its very simplicity the great ease which it has lent to all computations puts our arithmetic in the first rank of useful inventions and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of archimedes and apollonius two of the greatest minds produced by antiquity thanks for making us one of the best communities on youtube see you next episode of mind your decisions where we solve the world's problems one video at a time