140 LearnersLast updated on May 26th, 2025
Square Root of 401
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.
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.
Mistake 1
Forgetting about the Negative Square Root
It is important to be aware that a number has both positive and negative square roots. We typically focus on the positive square root, as it is often the required value.
For example, √401 ≈ 20.02498, but -20.02498 is also a square root that should not be forgotten.
Square Root of 401 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √401?
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.
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?
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.
Problem 3
Calculate √401 × 5.
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.
Problem 4
What will be the square root of (401 + 4)?
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.
Problem 5
Find the perimeter of a rectangle if its length ‘l’ is √401 units and the width ‘w’ is 40 units.
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.
FAQ on Square Root of 401
1.What is √401 in its simplest form?
2.Is 401 a prime number?
3.Calculate the square of 401.
4.How can I approximate the square root of a non-perfect square?
5.Why is √401 considered an irrational number?
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.
