Square Root of 728
2026-02-28 12:40 Diff
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Last updated on August 5, 2025

If a number is multiplied by itself, the result is a square. The inverse of the square is a square root. The square root is used in various fields such as vehicle design, finance, etc. Here, we will discuss the square root of 728.

What is the Square Root of 728?

The square root is the inverse of the square of a number. 728 is not a perfect square. The square root of 728 can be expressed in both radical and exponential forms. In radical form, it is expressed as √728, whereas in exponential form it is (728)^(1/2). √728 ≈ 26.9737, which is an irrational number because it cannot be expressed as a ratio of two integers (p/q, where q ≠ 0).

Finding the Square Root of 728

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, methods like the long division method and approximation method are used. Let us now learn the following methods:

  • Prime factorization method
  • Long division method
  • Approximation method

Square Root of 728 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 728 is broken down into its prime factors.

Step 1: Finding the prime factors of 728

Breaking it down, we get 2 × 2 × 2 × 91 = 2^3 × 7 × 13

Step 2: Now we found the prime factors of 728. The second step is to make pairs of those prime factors. Since 728 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating √728 using prime factorization alone is impractical.

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Square Root of 728 by Long Division Method

The long division method is particularly used for non-perfect square numbers. 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 728, we need to group it as 28 and 7.

Step 2: Now we need to find n whose square is less than or equal to the first group (7). We can say n is ‘2’ because 2 × 2 = 4, which is less than or equal to 7. Now the quotient is 2; after subtracting 4 from 7, the remainder is 3.

Step 3: Now, bring down 28 to make it 328, which is the new dividend. Double the current quotient (2) to get 4, which becomes our new divisor (4_).

Step 4: Find a digit to fill in the blank (4_) such that the product of this new number and the same digit is less than or equal to 328.

Step 5: Continue this process until you achieve the desired precision. Using this method, you will find that √728 ≈ 26.97.

Square Root of 728 by Approximation Method

The approximation method is another method for finding the square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 728 using the approximation method.

Step 1: Find the closest perfect squares around 728. The smallest perfect square less than 728 is 676 (26^2), and the largest perfect square greater than 728 is 729 (27^2). Thus, √728 falls between 26 and 27.

Step 2: Apply the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) (728 - 676) / (729 - 676) = 52/53 ≈ 0.9811 Add this decimal to the smaller square root: 26 + 0.9811 = 26.9811 Therefore, the square root of 728 is approximately 26.9811.

Common Mistakes and How to Avoid Them in the Square Root of 728

Students often make mistakes while finding square roots, such as forgetting about the negative square root and skipping steps in the long division method. Let us look at a few common mistakes 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 √728?

Okay, lets begin

The area of the square is 728 square units.

Explanation

The area of the square = side^2.

The side length is given as √728.

Area of the square = side^2 = √728 × √728 = 728.

Therefore, the area of the square box is 728 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 728 square feet is built. If each of the sides is √728, what will be the square feet of half of the building?

Okay, lets begin

364 square feet

Explanation

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

Dividing 728 by 2 gives us 364.

So, half of the building measures 364 square feet.

Well explained 👍

Problem 3

Calculate √728 × 5.

Okay, lets begin

134.87

Explanation

The first step is to find the square root of 728, which is approximately 26.97.

The second step is to multiply 26.97 by 5.

So, 26.97 × 5 ≈ 134.87.

Well explained 👍

Problem 4

What will be the square root of (728 + 1)?

Okay, lets begin

The square root is 27.

Explanation

To find the square root, we need to find the sum of (728 + 1). 728 + 1 = 729, and the square root of 729 is 27.

Therefore, the square root of (728 + 1) is 27.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √728 units and the width ‘w’ is 38 units.

Okay, lets begin

The perimeter of the rectangle is 129.94 units.

Explanation

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

Perimeter = 2 × (√728 + 38) ≈ 2 × (26.97 + 38) ≈ 2 × 64.97 ≈ 129.94 units.

Well explained 👍

FAQ on Square Root of 728

1.What is √728 in its simplest form?

The prime factorization of 728 is 2 × 2 × 2 × 7 × 13, so the simplest form of √728 is √(2^3 × 7 × 13).

2.Mention the factors of 728.

Factors of 728 are 1, 2, 4, 7, 8, 13, 14, 26, 28, 52, 56, 91, 104, 182, 364, and 728.

3.Calculate the square of 728.

We get the square of 728 by multiplying the number by itself: 728 × 728 = 529,984.

4.Is 728 a prime number?

5.728 is divisible by?

728 has many factors; these are 1, 2, 4, 7, 8, 13, 14, 26, 28, 52, 56, 91, 104, 182, 364, and 728.

Important Glossaries for the Square Root of 728

  • Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the inverse operation is the square root, √16 = 4.
  • Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
  • Long division method: A mathematical procedure used to find the square root of numbers, especially non-perfect squares, by breaking down the calculation into simpler steps.
  • Approximation method: A technique to estimate the value of a square root by using nearby perfect squares to find a closer value.
  • Prime factorization: The process of expressing a number as a product of its prime factors.

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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.