How to find Square Root of 729
Square of 27:
- In mathematics, finding of square of any number is mostly easy because when we multiply the same number with itself then we will get the square of that number.
For example:
- Let us suppose we have to find the square of any number say X, then we multiply X by itself i.e. X and we will get its square as Y. It can be written as (X)2 = X*X= Y
- In similar way we find the square of 27
- To find the square of 27,we multiply 27 by the number itself i.e. by 27 and we write it as follows. (729)2 = 27*27= 729
Square root of 729:
- Now, in reverse manner if we have to find the square root of Y. The square root of Y is that single value which when multiplied with itself gives the value Y.
- That means, √Y = √(X*X) = X
Where √ is the symbol named as radical.
For example:
- The square root of 27 can be written as,
√729= √ (27*27) = 27
Where √ is the symbol which is called as radical sign.
- In short, we remember square of 27 and square root of 729 as
Note:
- Every positive real number has two roots.
- The square of any negative number is always the positive number.
For example:
- 729 is the positive perfect square which has two roots +27 and -27 also.
- But, the positive square root value is taken mostly which is called as principal square root or non-negative square root.
- Hence, √729 = √(-27)*(-27) = -27 and √729 = √(27)*(27) = 27
Similarly,
- (-27)*(-27) = (-27)2 = +729 and (+27)*(+27) = (+27)2 = 729
Methods to find square root of perfect square like 729:
There are many methods to find the square root of perfect squares out of which we see the following method in detail.
- Repeated Subtraction Method
- Prime factorization method
Repeated Subtraction Method:
- In repeated subtraction method, we have to subtract the consecutive odd numbers starting from 1, from the perfect square number whose square root we have to find.
- e. to find square root of 729, first we subtract 1 from it. 729– 1 =728
- Then next odd number is 3, so we have to subtract it from 728. 728– 3 = 725
- In this way, we subtract the consecutive odd numbers from the corresponding values obtained after subtraction continuously till we get final value as 0.
- And the value of number of odd numbers required to get 0 is the required square root.
For example:
- We find the square root of 729 by repeated subtraction method as follows:
729 – 1 =728
728 – 3 = 725
725– 5 = 720
720– 7 = 713
713– 9 = 704
704 – 11 = 693
693 – 13 = 680
680 – 15 = 665
665 – 17 = 648
648– 19 = 629
629 – 21 = 608
608 – 23 = 585
585 – 25 – 560
560 – 27 = 533
533 – 29 = 504
504– 31 = 473
473 – 33 = 440
440 – 35 = 405
405 – 37 = 368
368 – 39 = 329
329 – 41 = 288
288 – 43 = 245
245– 45 = 200
200 – 47 = 153
153– 49 = 104
104 – 51 = 53
53 – 53 = 0
- Thus, here the total odd numbers used are 1, 3, 5, 7, 9, 11, 13, 15 ,17,19,21,23,25,27,29 ,31,33,35 37,39 ,41,43,45,47 49,51,53which are 27 in numbers.
- Hence, the square root of 729 by repeated subtraction method is 27.
Prime Factorization method:
- In prime factorization method, we have to divide the perfect square number whose square root we have to find by prime number starting from 2, 3, 5… and so on till we get the remainder as 1.
- Initially we have to divide by prime number 2, if that number is not divisible by 2 then we have to take next prime number i.e. 3 and the process will be continued till we get remainder as 1.
- Finally, we have to make pairs of the prime numbers taken in the form of multiplication and then we have to take its square root.
For example:
- Following is the process to find the square root of 729 by prime factorization method.
- As 729 is odd number hence it must be divisible by only prime number 3
729÷3=243
243÷3=81
81÷3=27
27÷3=9
9÷3= 3
3÷3=1
- Thus, the prime number 3 used to get remainder as 1 are 3,3,3,3,3,3
Thus, 729= 3*3*3*3*3*3= 3^2*3^2*3^2
And 729= (3*3*3*3*3*3)
- By taking square root on both sides, we get
√729 = √(3*3*3*3*3*3) = √(3*3)√(3*3)√(3*3)
√729 = (3*3*3)=27
- Thus, we found the square root of 729 as 27 by using prime factorization method.
Multiple choice questions:
1) using prime factorization method the square root of 729 is ——–
a) +27 and -27
b) +7 and -7
c)+72 and -72
d) none of these
Ans: a) +27 and -27
2) when we multiply the same number with itself then we will get the ——–of that number.
a) cube
b) square
c) cube root
d) none of these
Ans: b) square
3) In repeated subtraction method, we have to subtract the consecutive odd numbers starting from —–
a) 1
b) 3
c) 5
d)9
Ans: a) 1