Finding square root of a number by Prime Factorisation | Part 1/2

追加: 2026-05-27

[00:00:02]Let us find the square root of 729 using the prime factorization method.

[00:00:10]As per the name of this method, we can guess that we have to do the prime factorization of the number.

[00:00:19]In this method, we divide the number by prime numbers until we get a prime number or one.

[00:00:30]For this, we will draw a figure similar to the one on the screen. In the second column of the top row, we will write the number that we wish to factoriize.

[00:00:42]Now, let's get started. Find the smallest prime number that divides 729 completely.

[00:00:52]That is three.

[00:00:54]Write three in the first column of the first line.

[00:00:59]Dividing 729 by 3 gives us 243.

[00:01:06]Write it below 729.

[00:01:10]Now find the smallest prime number that divides 243 completely.

[00:01:17]That is three.

[00:01:19]Write three in the first column of the second line. Dividing 243 by 3 gives 81.

[00:01:29]Write it below 243.

[00:01:33]Now find the smallest prime number that divides 81 completely.

[00:01:40]That is 3.

[00:01:43]Write three in the first column of the third line.

[00:01:47]Dividing 81 by 3 gives the result 27.

[00:01:53]Write it below 81.

[00:01:56]27 is not a prime number. 3 is the prime factor of 27. Write it in its designated place.

[00:02:05]Dividing 27 by 3 gives us 9.

[00:02:10]Write 9 under 27.

[00:02:13]We know that the square of 3 is equal to 9. So we will write three in the first column of the fifth row and also under 9.

[00:02:25]Now we get the number three in the last row of the second column which is a prime number which has only two factors.

[00:02:34]The numbers 1 and three which we mark like this.

[00:02:41]All the prime factors of the number 729 are written in the first column and number one is the remainder in the last row of the second column indicating that the process is over.

[00:02:57]We can write the number 729 as the product of its prime factors.

[00:03:05]Now we will see how many prime numbers are in pairs in this factorization.

[00:03:12]There are three pairs of number three.

[00:03:15]By the laws of exponents, we can write this multiplication of factors as we get 729 as the square of 27. That is we can say that the square root of 729 is 27. Thus, here we get the square root of the number 729.