Square Root of 731
2026-02-28 13:56 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 fields like vehicle design, finance, etc. Here, we will discuss the square root of 731.

What is the Square Root of 731?

The square root is the inverse of the square of a number. 731 is not a perfect square. The square root of 731 is expressed in both radical and exponential form. In the radical form, it is expressed as √731, whereas in the exponential form it is expressed as (731)¹/². √731 ≈ 27.04163, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

Finding the Square Root of 731

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where 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 731 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 731 is broken down into its prime factors.

Step 1: Finding the prime factors of 731 731 is a prime number itself. Hence, the prime factorization of 731 is 731.

Step 2: Since 731 is not a perfect square, calculating the square root using prime factorization directly is not feasible.

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Square Root of 731 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, group the numbers from right to left. In the case of 731, group it as 31 and 7.

Step 2: Find n whose square is closest to or less than 7. Here, n is 2 because 2² = 4 is less than 7. The quotient is 2, and the remainder is 7 - 4 = 3.

Step 3: Bring down 31, making the new dividend 331. Add the old divisor (2) with itself to get 4, the start of the new divisor.

Step 4: Determine n such that 4n × n ≤ 331. Trying n = 8, we find 48 × 8 = 384, which is too large. Using n = 7, 47 × 7 = 329.

Step 5: Subtract 329 from 331 to get a remainder of 2. The quotient now is 27.

Step 6: Add a decimal point and two zeros to the dividend, making it 200.

Step 7: The new divisor is 54 (27 doubled and plus a digit to test). Continue refining until you approximate to two decimal places.

So, the square root of √731 ≈ 27.04.

Square Root of 731 by Approximation Method

The approximation method is another method for finding 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 731 using the approximation method.

Step 1: Find the closest perfect squares to √731. The smallest perfect square below 731 is 729 (27²), and the largest is 784 (28²). √731 falls between 27 and 28.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). For 731, (731 - 729) / (784 - 729) = 2 / 55 ≈ 0.036. Adding this to 27, we get 27 + 0.036 = 27.036.

So, the square root of 731 is approximately 27.04.

Common Mistakes and How to Avoid Them in the Square Root of 731

Students make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in the long division method. Now let us look at a few of those mistakes in detail.

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Problem 1

Can you help Max find the area of a square box if the side length is given as √731?

Okay, lets begin

The area of the square is 731 square units.

Explanation

The area of a square = side².

The side length is given as √731.

Area of the square = (√731) × (√731) = 731.

Therefore, the area of the square box is 731 square units.

Well explained 👍

Problem 2

A square-shaped land measuring 731 square feet is surveyed; if each of the sides is √731, what will be the square feet of half of the land?

Okay, lets begin

365.5 square feet

Explanation

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

Dividing 731 by 2, we get 365.5.

So, half of the land measures 365.5 square feet.

Well explained 👍

Problem 3

Calculate √731 × 5.

Okay, lets begin

135.21

Explanation

First, find the square root of 731, which is approximately 27.04.

Then multiply 27.04 by 5. So, 27.04 × 5 = 135.20.

Well explained 👍

Problem 4

What will be the square root of (729 + 2)?

Okay, lets begin

The square root is approximately 27.04.

Explanation

To find the square root, calculate the sum of (729 + 2). 729 + 2 = 731, and then √731 ≈ 27.04.

Therefore, the square root of (729 + 2) is approximately 27.04.

Well explained 👍

Problem 5

Find the perimeter of a rectangle if its length ‘l’ is √731 units and the width ‘w’ is 10 units.

Okay, lets begin

The perimeter of the rectangle is approximately 94.08 units.

Explanation

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

Perimeter = 2 × (√731 + 10) = 2 × (27.04 + 10) = 2 × 37.04 = 74.08 units.

Well explained 👍

FAQ on Square Root of 731

1.What is √731 in its simplest form?

Since 731 is a prime number, √731 remains in its simplest form as it cannot be simplified further.

2.Is 731 a prime number?

Yes, 731 is a prime number as it has no divisors other than 1 and itself.

3.Calculate the square of 731.

We get the square of 731 by multiplying the number by itself: 731 × 731 = 534,361.

4.Is 731 divisible by any other numbers besides 1 and 731?

No, since 731 is a prime number, it is not divisible by any numbers other than 1 and 731.

5.What are the uses of square roots in real life?

Square roots are used in various fields, such as engineering, physics, statistics, and finance, to solve equations involving areas, velocities, and other measurements.

Important Glossaries for the Square Root of 731

  • Square root: A square root is the inverse operation of squaring a number. Example: If 5² = 25, then √25 = 5.
  • Irrational number: An irrational number cannot be written as a simple fraction. Its decimal form goes on forever without repeating. Example: √2, √731.
  • Prime number: A prime number is a number greater than 1 that has no divisors other than 1 and itself. Example: 2, 3, 5, 731.
  • Approximation: Approximation refers to finding a value that is close enough to the correct answer, usually with some degree of accuracy. Example: √731 ≈ 27.04.
  • Long division: A method used to divide large numbers and determine the quotient with a remainder, often used for calculating square roots of non-perfect squares.

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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.