Square Root of 729
2026-02-28 12:15 Diff
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Last updated on August 5, 2025

If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in the fields of vehicle design, finance, etc. Here, we will discuss the square root of 729.

What is the Square Root of 729?

The square root is the inverse of the square of the number. 729 is a perfect square. The square root of 729 is expressed in both radical and exponential form. In the radical form, it is expressed as √729, whereas (729)^(1/2) in the exponential form. √729 = 27, which is a rational number because it can be expressed in the form of p/q, where p and q are integers and q ≠ 0.

Finding the Square Root of 729

The prime factorization method is used for perfect square numbers like 729. The long-division method and approximation method can also be used but are not necessary. Let us now learn the following methods:

  • Prime factorization method
  • Long division method
  • Approximation method

Square Root of 729 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 729 is broken down into its prime factors.

Step 1: Finding the prime factors of 729

Breaking it down, we get 3 x 3 x 3 x 3 x 3 x 3: 3^6

Step 2: Now we found out the prime factors of 729. The second step is to make pairs of those prime factors. Since 729 is a perfect square, the digits of the number can be grouped in pairs. Therefore, calculating √729 using prime factorization is possible, and we get 3^3 = 27.

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Square Root of 729 by Long Division Method

The long division method is useful for verifying square roots. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, we need to group the numbers from right to left. In the case of 729, we need to group it as 29 and 7.

Step 2: Now we need to find n whose square is 7. We can say n as ‘2’ because 2 x 2 is less than or equal to 7. Now the quotient is 2, and after subtracting 4 from 7, the remainder is 3.

Step 3: Now bring down 29, making the new dividend 329. Add the old divisor with the quotient, 2 + 2 = 4, which will be our new divisor.

Step 4: The new divisor will be 4n. We need to find the value of n such that 4n x n ≤ 329. Let us consider n as 7; now 47 x 7 = 329. Step 5: Subtract 329 from 329; the difference is 0, and the quotient is 27.

So the square root of √729 is 27.

Square Root of 729 by Approximation Method

The approximation method can be used to verify the square roots, but it is not necessary for perfect squares like 729. We can directly find the square root of 729 as 27 using the prime factorization or long division method.

Common Mistakes and How to Avoid Them in the Square Root of 729

Students do make mistakes while finding the square root, like forgetting about the negative square root or skipping steps in the long division method. Now let us look at a few of those mistakes that students tend to make in detail.

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Problem 1

Can you help Max find the area of a square box if its side length is given as √729?

Okay, lets begin

The area of the square is 729 square units.

Explanation

The area of the square = side^2.

The side length is given as √729.

Area of the square = side^2 = √729 x √729 = 27 x 27 = 729.

Therefore, the area of the square box is 729 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 729 square feet is built; if each of the sides is √729, what will be the square feet of half of the building?

Okay, lets begin

364.5 square feet

Explanation

We can just divide the given area by 2 as the building is square-shaped.

Dividing 729 by 2, we get 364.5.

So half of the building measures 364.5 square feet.

Well explained 👍

Problem 3

Calculate √729 x 5.

Okay, lets begin

135

Explanation

The first step is to find the square root of 729, which is 27.

The second step is to multiply 27 with 5.

So 27 x 5 = 135.

Well explained 👍

Problem 4

What will be the square root of (625 + 104)?

Okay, lets begin

The square root is 27.

Explanation

To find the square root, we need to find the sum of (625 + 104). 625 + 104 = 729, and then √729 = 27.

Therefore, the square root of (625 + 104) is ±27.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √729 units and the width ‘w’ is 10 units.

Okay, lets begin

We find the perimeter of the rectangle as 74 units.

Explanation

Perimeter of the rectangle = 2 × (length + width).

Perimeter = 2 × (√729 + 10) = 2 × (27 + 10) = 2 × 37 = 74 units.

Well explained 👍

FAQ on Square Root of 729

1.What is √729 in its simplest form?

The prime factorization of 729 is 3 x 3 x 3 x 3 x 3 x 3, so the simplest form of √729 = 3^3 = 27.

2.Mention the factors of 729.

Factors of 729 are 1, 3, 9, 27, 81, 243, and 729.

3.Calculate the square of 729.

We get the square of 729 by multiplying the number by itself, that is 729 x 729 = 531441.

4.Is 729 a prime number?

5.729 is divisible by?

729 has several factors; those are 1, 3, 9, 27, 81, 243, and 729.

Important Glossaries for the Square Root of 729

  • Square root: A square root is the inverse of a square. Example: 6^2 = 36, and the inverse of the square is the square root that is √36 = 6.
  • Rational number: A rational number is a number that can be written in the form of p/q, where q is not equal to zero, and p and q are integers.
  • Perfect square: A number that is the square of an integer. For example, 729 is a perfect square because it is 27^2.
  • Prime factorization: Breaking down a number into its prime factors. For example, the prime factorization of 729 is 3 x 3 x 3 x 3 x 3 x 3.
  • Perimeter: The perimeter is the distance around a two-dimensional shape. For example, the perimeter of a rectangle is 2 × (length + width).

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.