Square Root of 371
2026-02-28 01:31 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 squaring a number is finding its square root. Square roots are used in various fields such as engineering, finance, and more. Here, we will discuss the square root of 371.

What is the Square Root of 371?

The square root is the inverse operation of squaring a number. 371 is not a perfect square. The square root of 371 can be expressed in both radical and exponential forms. In radical form, it is expressed as √371, whereas in exponential form it is written as 371^(1/2). The square root of 371 is approximately 19.26136, which is an irrational number because it cannot be expressed as a fraction of two integers.

Finding the Square Root of 371

For non-perfect square numbers like 371, methods such as the long division method and approximation method are employed, instead of the prime factorization method used for perfect squares. Let's explore these methods:

  • Long division method 
  • Approximation method

Square Root of 371 by Long Division Method

The long division method is commonly used for finding square roots of non-perfect squares. Here's how it's done step by step for 371:

Step 1: Group the numbers from right to left. For 371, consider it as 371.

Step 2: Find the largest number whose square is less than or equal to the group. Here, 5^2 = 25, and 6^2 = 36. Use 6 as the divisor because 36 is the closest perfect square less than 37. The quotient is 6, and the remainder is 1 (37 - 36 = 1).

Step 3: Bring down the next pair of digits, making the new dividend 171.

Step 4: Double the quotient (6), resulting in 12. Use 12 as the beginning of the new divisor, and find a digit n such that 12n * n is less than or equal to 171.

Step 5: The closest value for n is 4, since 124 * 4 = 496. Subtract 496 from 1710 to get the remainder 54.

Step 6: Add a decimal point to the quotient and pair two zeros to the remainder, making it 5400.

Step 7: Repeat the process with new dividends and divisors until the desired decimal precision is achieved.

The square root of 371 is approximately 19.26136.

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Square Root of 371 by Approximation Method

The approximation method provides a quick way to find the square root by identifying the closest perfect squares surrounding the number.

Step 1: Identify the perfect squares closest to 371.

The perfect squares around 371 are 361 (19^2) and 400 (20^2).

So, √371 is between 19 and 20.

Step 2: Use interpolation to approximate the decimal value.

The formula is:

(Given number - smaller perfect square) / (larger perfect square - smaller perfect square).

For 371, it is: (371 - 361) / (400 - 361) = 10 / 39 ≈ 0.256

Step 3: Add this fractional part to the lower square root: 19 + 0.256 ≈ 19.26

Thus, the square root of 371 is approximately 19.26.

Common Mistakes and How to Avoid Them in the Square Root of 371

Finding square roots can lead to common errors, such as ignoring the negative square root or misapplying methods. Let's explore some frequent mistakes and how to avoid them.

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Problem 1

Can you help Max find the area of a square box if its side length is given as √371?

Okay, lets begin

The area of the square is approximately 371 square units.

Explanation

The area of a square is calculated as side^2.

Given the side length is √371, the area is √371 * √371 = 371 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 371 square feet is built; if each of the sides is √371, what will be the square feet of half of the building?

Okay, lets begin

185.5 square feet

Explanation

Since the building is square-shaped, its total area is 371 square feet.

Half of the area is 371 / 2 = 185.5 square feet.

Well explained 👍

Problem 3

Calculate √371 × 5.

Okay, lets begin

96.31

Explanation

First, find the square root of 371, which is approximately 19.26.

Then multiply by 5: 19.26 × 5 = 96.31.

Well explained 👍

Problem 4

What will be the square root of (371 + 29)?

Okay, lets begin

The square root is approximately 20.

Explanation

First, find the sum (371 + 29) = 400.

Then, the square root of 400 is 20.

Therefore, the square root of (371 + 29) is ±20.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √371 units and the width ‘w’ is 12 units.

Okay, lets begin

We find the perimeter of the rectangle as approximately 62.52 units.

Explanation

The perimeter of a rectangle is calculated as 2 × (length + width).

Perimeter = 2 × (√371 + 12) ≈ 2 × (19.26 + 12) = 2 × 31.26 = 62.52 units.

Well explained 👍

FAQ on Square Root of 371

1.What is √371 in its simplest form?

The prime factorization of 371 does not result in any perfect square factors, so the simplest form of √371 remains √371.

2.Mention the factors of 371.

The factors of 371 are 1, 7, 53, and 371.

3.Calculate the square of 371.

The square of 371 is found by multiplying the number by itself: 371 × 371 = 137,641.

4.Is 371 a prime number?

371 is not a prime number, as it can be divided by 1, 7, 53, and 371.

5.371 is divisible by?

371 is divisible by 1, 7, 53, and 371.

Important Glossaries for the Square Root of 371

  • Square root: A square root is the inverse operation of squaring a number. For example, 5^2 = 25, and the square root of 25 is √25 = 5.
     
  • Irrational number: An irrational number is a number that cannot be written as a simple fraction, with non-repeating and non-terminating decimal expansion.
     
  • Principal square root: The principal square root is the non-negative square root of a number. For example, the principal square root of 25 is 5, not -5.
     
  • Perfect square: A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because it is 6^2.
     
  • Long division method: A method used to find the square root of non-perfect squares by performing a series of divisions and approximations.

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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.