Square Root of 683
2026-02-28 09:06 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 this process is finding the square root. Square roots are used in fields like vehicle design, finance, etc. Here, we will discuss the square root of 683.

What is the Square Root of 683?

The square root is the inverse of squaring a number. 683 is not a perfect square. The square root of 683 can be expressed in both radical and exponential forms. In radical form, it is expressed as √683, whereas in exponential form, it is (683)¹/². √683 ≈ 26.1108, which is an irrational number because it cannot be expressed as a ratio p/q, where p and q are integers and q ≠ 0.

Finding the Square Root of 683

The prime factorization method is suitable for perfect square numbers. However, for non-perfect squares like 683, methods like long division or approximation are used. Let's explore these methods: 

  • Prime factorization method 
  • Long division method 
  • Approximation method

Square Root of 683 by Prime Factorization Method

The prime factorization of a number is the product of its prime factors. Let's see how 683 is broken down into its prime factors.

Step 1: Finding the prime factors of 683 683 is a prime number itself, having no factors other than 1 and 683. Therefore, the prime factorization method is not applicable for finding its square root.

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Square Root of 683 by Long Division Method

The long division method is particularly used for non-perfect square numbers. Let's learn how to find the square root using this method, step by step.

Step 1: Group the numbers from right to left. For 683, we consider 683 as a single group.

Step 2: Find the largest number whose square is less than or equal to 683. This number is 26, since 26² = 676, which is less than 683.

Step 3: Subtract 676 from 683, leaving a remainder of 7.

Step 4: Bring down pairs of zeros to the right of the remainder.

Step 5: Double the divisor (26) to get 52, and find a digit x such that 52x * x is less than or equal to 700. Here, x = 1 works, as 521 * 1 = 521.

Step 6: Subtract 521 from 700, leaving 179.

Step 7: Bring down another pair of zeros, making it 17900.

Step 8: Continue the division process to get the desired precision. Thus, √683 ≈ 26.11.

Square Root of 683 by Approximation Method

The approximation method is a simpler way to find square roots. Let's use it to find the square root of 683.

Step 1: Identify the perfect squares closest to 683. The perfect squares are 676 (26²) and 729 (27²). Thus, √683 is between 26 and 27.

Step 2: Use linear interpolation: (683 - 676) / (729 - 676) = 7 / 53 ≈ 0.132

So, √683 ≈ 26 + 0.132 = 26.132

Common Mistakes and How to Avoid Them in the Square Root of 683

Students often make mistakes while finding square roots, such as ignoring negative roots or skipping steps in the long division method. Let's explore common mistakes in detail.

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Problem 1

Can you help Max find the area of a square if its side length is given as √683?

Okay, lets begin

The area of the square is approximately 683 square units.

Explanation

The area of a square is side².

Given the side length is √683

Area = (√683)² = 683 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 683 square feet is built; if each of the sides is √683, what will be the square feet of half of the building?

Okay, lets begin

341.5 square feet

Explanation

For a square-shaped building, dividing the total area by 2 gives the area of half the building.

683 / 2 = 341.5 square feet

Well explained 👍

Problem 3

Calculate √683 x 5.

Okay, lets begin

Approximately 130.554

Explanation

First, find the square root of 683, which is approximately 26.1108.

Then multiply by 5: 26.1108 x 5 ≈ 130.554

Well explained 👍

Problem 4

What will be the square root of (683 + 17)?

Okay, lets begin

The square root is approximately 26.419

Explanation

First, find the sum of 683 + 17 = 700.

Then, find the square root of 700, which is approximately 26.419.

Well explained 👍

Problem 5

Find the perimeter of a rectangle if its length ‘l’ is √683 units and the width ‘w’ is 38 units.

Okay, lets begin

The perimeter of the rectangle is approximately 128.2216 units.

Explanation

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

Perimeter = 2 × (√683 + 38) = 2 × (26.1108 + 38) ≈ 2 × 64.1108 = 128.2216 units.

Well explained 👍

FAQ on Square Root of 683

1.What is √683 in its simplest form?

Since 683 is a prime number, √683 cannot be simplified further and remains in its radical form as √683.

2.Is 683 a Prime Number?

Yes, 683 is a prime number because it has no divisors other than 1 and 683 itself.

3.Calculate the square of 683.

We find the square of 683 by multiplying the number by itself: 683 x 683 = 466,489.

4.What are the factors of 683?

Since 683 is a prime number, its only factors are 1 and 683.

5.Is 683 divisible by any number other than 1 and itself?

No, 683 is not divisible by any number other than 1 and itself, as it is a prime number.

Important Glossaries for the Square Root of 683

  • Square root: The square root is the inverse operation of squaring a number. For example, 4² = 16, and the square root of 16 is √16 = 4.
  • Irrational number: An irrational number cannot be expressed as a simple fraction p/q, where p and q are integers, and q ≠ 0.
  • Prime number: A prime number is a number greater than 1 that has no divisors other than 1 and itself.
  • Linear interpolation: A method used to estimate values between two known values on a line or curve.
  • Long division method: A step-by-step method used to divide numbers and find square roots, especially useful for 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.