Square Root of 401
2026-02-28 13:56 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 fields such as vehicle design and finance. Here, we will discuss the square root of 401.

What is the Square Root of 401?

The square root is the inverse of the square of a number. 401 is not a perfect square. The square root of 401 is expressed in both radical and exponential form. In radical form, it is expressed as √401, whereas in exponential form, it is (401)^(1/2). √401 ≈ 20.02498, which is an irrational number because it cannot be expressed in the form p/q, where p and q are integers and q ≠ 0.

Finding the Square Root of 401

The prime factorization method can be used for perfect square numbers. However, for non-perfect square numbers, 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 401 by Prime Factorization Method

Prime factorization involves expressing a number as the product of its prime factors. Since 401 is a prime number, it cannot be broken down into other prime factors. Therefore, calculating the square root of 401 using prime factorization is not possible.

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Square Root of 401 by Long Division Method

The long division method is particularly useful for non-perfect square numbers. Here's how to find the square root of 401 using the long division method, step by step:

Step 1: Group the digits of 401 from right to left. In the case of 401, we have one group of three digits: 401.

Step 2: Find the largest number whose square is less than or equal to 401. The closest perfect square is 400, and its square root is 20.

Step 3: Subtract the square of 20 from 401, which gives a remainder of 1. The quotient is 20.

Step 4: Bring down a pair of zeroes to make the dividend 100.

Step 5: Double the quotient (20) which gives us a new divisor of 40.

Step 6: Find a digit n such that 40n × n is less than or equal to 100. The suitable n is 2, as 402 × 2 = 80.

Step 7: Subtract 80 from 100, leaving a remainder of 20.

Step 8: Continue the process by bringing down more pairs of zeroes and repeating the steps above to get more decimal places in the quotient.

The square root of 401 is approximately 20.02498.

Square Root of 401 by Approximation Method

The approximation method is another way to find square roots. It is an easy method for estimating the square root of a number. Here's how to approximate the square root of 401:

Step 1: Identify the closest perfect squares around 401.

The closest are 400 (20²) and 441 (21²).

√401 falls between 20 and 21.

Step 2: Use the formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

For 401, this is (401 - 400) / (441 - 400) = 1 / 41 ≈ 0.02439.

Step 3: Add this result to the smaller root: 20 + 0.02439 ≈ 20.02439.

Therefore, the square root of 401 is approximately 20.02439.

Common Mistakes and How to Avoid Them in the Square Root of 401

Students often make mistakes while finding square roots, such as forgetting about the negative square root or skipping long division steps. Let's examine a few common mistakes in more 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 √401?

Okay, lets begin

The area of the square is approximately 401 square units.

Explanation

The area of the square = side².

The side length is given as √401.

Area of the square = (√401)² = 401.

Therefore, the area of the square box is approximately 401 square units.

Well explained 👍

Problem 2

A square-shaped garden measures 401 square feet. If each side is √401 feet long, what will be the area of half of the garden?

Okay, lets begin

200.5 square feet

Explanation

Since the garden is square-shaped, divide the total area by 2. 401 ÷ 2 = 200.5.

So, half of the garden measures 200.5 square feet.

Well explained 👍

Problem 3

Calculate √401 × 5.

Okay, lets begin

Approximately 100.12

Explanation

First, find the square root of 401, which is approximately 20.02498.

Then multiply it by 5. 20.02498 × 5 ≈ 100.12.

Well explained 👍

Problem 4

What will be the square root of (401 + 4)?

Okay, lets begin

The square root is approximately 20.4206.

Explanation

To find the square root, first calculate the sum: 401 + 4 = 405.

Then find the square root: √405 ≈ 20.1246.

Therefore, the square root of (401 + 4) is approximately 20.1246.

Well explained 👍

Problem 5

Find the perimeter of a rectangle if its length ‘l’ is √401 units and the width ‘w’ is 40 units.

Okay, lets begin

The perimeter of the rectangle is approximately 120.05 units.

Explanation

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

Perimeter = 2 × (√401 + 40) ≈ 2 × (20.02498 + 40) ≈ 2 × 60.02498 ≈ 120.05 units.

Well explained 👍

FAQ on Square Root of 401

1.What is √401 in its simplest form?

Since 401 is a prime number, the simplest form of √401 is simply √401.

2.Is 401 a prime number?

Yes, 401 is a prime number, as it has only two factors: 1 and 401.

3.Calculate the square of 401.

To find the square of 401, multiply the number by itself: 401 × 401 = 160,801.

4.How can I approximate the square root of a non-perfect square?

To approximate the square root, identify the closest perfect squares and use them to estimate the root of the non-perfect square.

5.Why is √401 considered an irrational number?

√401 is irrational because it cannot be expressed as a fraction of two integers.

Important Glossaries for the Square Root of 401

  • Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, 4² = 16, so √16 = 4.
     
  • Irrational number: An irrational number cannot be written as a simple fraction, as it has a non-repeating and non-terminating decimal expansion.
     
  • Prime number: A prime number has only two factors—1 and itself.
     
  • Approximation: This refers to finding a value that is close to but not exactly equal to the actual value.
     
  • Long division method: A technique used to divide large numbers and can be adapted to find 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.