Square Root of 51
2026-02-28 11:57 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 various fields such as mathematics, physics, and engineering. Here, we will discuss the square root of 51.

What is the Square Root of 51?

The square root is the inverse of squaring a number. The number 51 is not a perfect square, and its square root is expressed in both radical and exponential forms.

In radical form, it is expressed as √51, whereas in exponential form it is (51)^(1/2). The square root of 51 is approximately 7.14143, which is an irrational number because it cannot be expressed as a fraction of two integers.

Finding the Square Root of 51

The prime factorization method is typically used for perfect square numbers. For non-perfect square numbers, the long-division method and approximation method are more suitable. Let's explore the following methods:

  • Long division method
  • Approximation method

Square Root of 51 by Long Division Method

The long division method is particularly useful for non-perfect squares. It involves finding the closest perfect square number to the given number. Let's find the square root of 51 using this method, step by step:

Step 1: To begin, we group the numbers from right to left. For 51, we have a single group: 51.

Step 2: Find the largest number whose square is less than or equal to 51. The number is 7, since 7 × 7 = 49. The quotient is 7, and after subtracting 49 from 51, the remainder is 2.

Step 3: Bring down a pair of zeros, making the new dividend 200.

Step 4: Double the quotient (7), resulting in 14, which will be part of our new divisor.

Step 5: Find the largest digit x such that 14x × x is less than or equal to 200. In this case, x is 1, since 141 × 1 = 141.

Step 6: Subtract 141 from 200, leaving a remainder of 59.

Step 7: Since the remainder is less than the divisor, add a decimal point and bring down a pair of zeros, making the new dividend 5900.

Step 8: Continue this process to get a more precise decimal value.

The square root of √51 is approximately 7.14.

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Square Root of 51 by Approximation Method

The approximation method offers an easy way to estimate square roots. Here's how to approximate the square root of 51:

Step 1: Identify the perfect squares near 51. The closest perfect squares are 49 (7^2) and 64 (8^2). Thus, √51 is between 7 and 8.

Step 2: Use linear interpolation: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) (51 - 49) / (64 - 49) = 2 / 15 ≈ 0.133

Step 3: Add this decimal to the smaller perfect square's root: 7 + 0.133 = 7.133

So, the approximate square root of 51 is 7.133.

Common Mistakes and How to Avoid Them in the Square Root of 51

Students often make mistakes when calculating square roots, such as ignoring the negative square root or skipping steps in the long division method. Let's examine some 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 √51?

Okay, lets begin

The area of the square is approximately 51 square units.

Explanation

The area of a square is calculated as side². Given the side length is √51:

Area = (√51)² = 51.

Therefore, the area of the square box is approximately 51 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 51 square feet is built; if each of the sides is √51, what will be the square feet of half of the building?

Okay, lets begin

25.5 square feet

Explanation

To find half of the building's area, simply divide the total area by 2. 51 / 2 = 25.5.

Thus, half of the building measures 25.5 square feet.

Well explained 👍

Problem 3

Calculate √51 x 5.

Okay, lets begin

Approximately 35.70715

Explanation

First, find the square root of 51, which is approximately 7.14143.

Then multiply by 5: 7.14143 x 5 ≈ 35.70715.

Well explained 👍

Problem 4

What will be the square root of (45 + 6)?

Okay, lets begin

The square root is 7.

Explanation

First, find the sum of 45 + 6 = 51.

Then, sqrt(51) ≈ 7.14143.

Therefore, the square root is approximately 7.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √51 units and the width ‘w’ is 10 units.

Okay, lets begin

Approximately 34.28286 units.

Explanation

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

Perimeter = 2 × (√51 + 10) ≈ 2 × (7.14143 + 10) ≈ 2 × 17.14143 ≈ 34.28286 units.

Well explained 👍

FAQ on Square Root of 51

1.What is √51 in its simplest form?

The prime factorization of 51 is 3 × 17, so the simplest radical form of √51 is √(3 × 17).

2.What are the factors of 51?

The factors of 51 are 1, 3, 17, and 51.

3.Calculate the square of 51.

We find the square of 51 by multiplying it by itself: 51 × 51 = 2601.

4.Is 51 a prime number?

No, 51 is not a prime number, as it has more than two factors (1, 3, 17, and 51).

5.51 is divisible by which numbers?

51 is divisible by 1, 3, 17, and 51.

Important Glossaries for the Square Root of 51

  • Square root: A square root is a value that, when multiplied by itself, gives the original number. Example: 4² = 16, and the square root of 16 is √16 = 4.
     
  • Irrational number: An irrational number cannot be written as a simple fraction; it has a non-repeating, non-terminating decimal expansion.
     
  • Non-perfect square: A number that does not have an integer as its square root. For example, 51 is a non-perfect square.
     
  • Long division method: A step-by-step method used to find the square root of non-perfect squares more accurately.
     
  • Decimal approximation: Estimating the value of a number by finding its decimal representation, particularly useful for irrational numbers.

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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.