Square Root of 729

Square root of 729 is 27. It is represented as √729 = 27. The square root of any number is an integer (can be positive or negative) which when multiplied by itself results in the given number. Let us consider the example of the square root of 729. The solution is 27 because 27 times 27 gives the number “729”. √729 can be either positive or negative. i.e 27 × 27 is 729 or – 27 × – 27 is also 729. Since 729 is a perfect square, the square root of 729 can be determined using the prime factorisation method, long division method, simplified method and repeated subtraction method. 

The exponential form of the square root of 729 is 7291/2 and the solution will be 272. The representation of the square root of 729 is √729, where ‘√’ is termed as the radical and 729 as the radicand. The square root of numbers that do not have an integer as a solution forms an irrational number. A detailed explanation of square roots can be obtained from here: Square Root.

What is the Square root of 729?

The square root of 729 is 27. In other words, the square of 27 is 729. i.e 27 × 27 is 729. 

How to Find the Square root of 729?

There are three methods to find the Square root of 729:

• Prime Factorisation method
• Long Division method
• Repeated Subtraction method

Square root of 729 by Prime Factorisation Method

In the Prime Factorisation method, the given number is expressed as the product of prime numbers and by grouping 2 same numbers. If the grouping of 2 same numbers is not possible, then the number is not a perfect square number. Or, it does not have a square root.

Given number, 729 will be expressed as;

729 = 3 × 3 × 3 × 3 × 3 × 3

Grouping the same numbers in pairs

 Group 1 = 3 × 3 

 Group 2 = 3 × 3 

 Group 3 = 3 × 3 

Considering the groups, square root of 729 = 3 × 3 × 3 = 27

Therefore square root of 729 = 27.

Square root of 729 by Long Division Method

In the long division method, the given numbers are paired in groups, starting from the right side. Let us understand the long division method, in detailed steps;

Step 1:  Grouping the given number into pairs

Given number is 729, grouping it as 7 and 29.

Step 2: Consider the first number, which is 7.

Let us find a square number that divides 7

i.e 1 × 1 = 1 

2 × 2 = 4 

3 × 3 = 9

The square number that is to be considered for dividing 7 will be 2 x 2

Step 3: Dividing 7 by 2

Step 4: Continue the division using the next number 29

The first number to be used as next divisor is 2 × 2 = 4 (Divisor of first division multiplied by quotient of first division)

Now considering a two digit number starting with 4 to divide 329 will be:

41 × 1 = 41

42 × 2 = 84

43 × 3 = 129

44 × 4 = 176

45 × 5 = 225

46 × 6 = 276

47 × 7 = 329

Continue the division as

Since the Division is complete with the remainder as zero, the quotient becomes the square root of the given number. 

Therefore the Square root of 729 is 27.

Square root of 729 by Repeated Subtraction Method.

In the repeated subtraction method, the given number is subtracted repeatedly by odd numbers till the final solution is zero. The total number of times the subtraction is performed becomes the square root. Please note that, if the given number is not a perfect square number, then the result 0 is not obtained.

For the given number 729, steps for repeated subtraction are:

Since, the result zero is obtained in the 27th Step, the square root of 729 is 27.

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Solved Examples

1. Find the square root of 729 by pairing factors?

The factors of 729 are 1, 3, 9, 27, 81, 243 and 729. The pairs are: 

1 × 729, 3 × 243, 9 × 81, 27 x 27. 

Therefore the square root of 729 is 27 using pairing factors 27 and 27. 

2. What is the square of 29?

The square of 29 is nothing but 29 times 29 and that is 841.