 Hello and welcome to the session. In this session we will discuss square roots. We know that inverse operation of addition is subtraction and inverse operation of multiplication is division. So we say that finding the square root is the inverse operation of squaring. Like we know that 4 square is equal to 16. So we say square root of 16 is 4 and we use this symbol to denote square root. So we can say that square root of 16 is 4 in this way. Next we discuss finding square root through repeated subtraction. As we know that sum of first n odd natural numbers is n square. Thus we say that every square number can be expressed as a sum of successive odd natural numbers starting from 1. Let's try and find out square root of 25 through repeated subtraction. To start with this we subtract successive odd numbers starting from 1 from the number 25. Like in the first step we subtract 1 from 25 and we get 24. Then in the second step we subtract the next odd number that is 3 from 24. So this gives us 21. Then we have 21 minus 5. Since 5 is the next odd number this gives us 16. Then in the next step we have 16 minus 7 that is 7 is the next odd number. So this gives us 9. Then in the fifth step we have 9 minus the next odd number which is 9. This gives us 0. Now since we have got 0 so we stop here. Now as you can see we have got 0 at the fifth step thus we say that square root of 25 would be equal to 5. Now next we have finding square root through prime factorization. Consider the number 4 its prime factorization is given as 2 into 2. Now let's consider 4 square that is 16. Its prime factorization is given as 2 into 2 into 2 into 2. You can easily observe that each prime factor in the prime factorization of the square of a number occurs twice. The number of times it occurs in the prime factorization of the number itself. Let's try and find out square root of 400 through prime factorization. First let's see what is the prime factorization of the number 400. This is equal to 2 into 2 into 2 into 2 into 5 into 5. Now each factor appears 2 times in the prime factorization of this number. Now let's make pairs of these prime factors. This is one pair, this is the other pair and this is the third pair. Now to find the square root of 400 we take out one factor from each pair. So from the first pair we have a 2 multiplied by a 2 from the second pair multiplied by a 5 from the third pair. So this gives us square root of 400 is equal to 20. This is how we find the square root of a number through prime factorization. Next we have finding square root by division method. Now sometimes when we are given large numbers we cannot find its square root through prime factorization so we use the division method to do so. If we have that a perfect square is of n digits then its square root will have n upon 2 digits if n is even or n plus 1 upon 2 digits if n is odd. Let's consider the number 400. Now as you can see there are 3 digits in this number so we say that n is equal to 3 here and 3 is odd. So we say that the number of digits in square root of 400 will be equal to n plus 1 upon 2 that is equal to 2. And as we have already found out that square root of 400 is 20 that is it has 2 digits. Let's try and find out square root of the number 2304 using the division method. Let's do it step by step. In the first step consider the number and replace the bar over every pair of digits starting from the digit at once place in this way and in case if the number of digits in the number is odd then the left most single digit will also have a bar. Then in the next step we find the largest number whose square is less than or equal to the number under the extreme left bar that is 23. We have 23 is less than 5 square and it is greater than 4 square. Then we take this number 4 in this case as the divisor and the question with the number under the extreme left bar that is 23 as the dividend and we divide them that is we put here 4 and here also 4 we get here 16 and the remainder that we get is 7. Then in the next step we bring down the number under the next bar that is 0 4 in this case to the right of the remainder 7 that we have got. So in this case we get that a new dividend is 704. Then in the next step we double the divisor and enter it with a blank on its right. Now double or 4 would be 8 so we write here 8 with a blank to its right. Then in the next step we need to guess the largest possible digit to fill this blank and the digit in this blank will also be the digit in the question such that when we multiply the new divisor to the new question the product is less than or equal to the dividend like in this case it would be 8 that is we have 88 into 8 is equal to 704. Now we got the remainder as 0. Now we have got the remainder as 0 and there are no digits left in the given number. So we stop here and we say that square root of the number 2304 is 48. This is how we can find the square root of a number by division method. We can also estimate the number of digits in the square root of a perfect square number by using these bars that is we have that the number of digits in the square root of a number is equal to the number of bars in the number like as you can see in the given number 2304 we have got two bars so we say that there would be two digits in the square root of this number and we have already found out the square root of this number which is 48 that is it has two digits. Next we try finding the square roots of decimals. Consider the decimal number 31.36 we need to find the square root of this number let's start doing this. Consider the number now we put the bars on the integral part of this number that is 31 in this case in the usual manner as we do. So we got this one bar over the integral part and in the decimal number we place the bars on every pair of digits beginning with the first decimal place so we got one bar over here. Now we proceed in the same manner that is we have 31 is less than 6 square and it is greater than 5 square. We take this number 5 as the divisor and the number under the left most bar that is 31 as the dividend and the number in the question also has to be the same so we divide them so we get here 6 as the remainder. Then in the next step we write the number under the next bar that is 36 in this case to the right of this remainder 6 that we have got. So we have got 636 is the new dividend then in the next step we double the divisor and enter it with a blank on its right that is double of 5 would be 10 and we put a blank over here now since 36 is the decimal part so we put a decimal point in the question in this way then we know that 106 into 6 is 636 so we put here 6 in the blank and in the question also we put 6 now we get the remainder as 0 so we get this question as the square root of the given number that is square root of 31.36 is 5.6 while putting the bars over the decimal part of a number we start from the decimal and we move towards the right we need to remember this point while putting the bars in the decimal number. Next we have estimating square root suppose we need to estimate the square root of the number 80 let's see how we do this now this number 80 is less than 81 and it is greater than 64 then we have square root of 80 is less than square root of 81 and is greater than square root of 64 that is square root of 80 is less than 9 and it is greater than 8 now since 81 is much closer to 80 than 64 thus we say that square root of 80 is approximately 9 this is how we can estimate the square root of any number this completes the session hope you have understood the concept of square root