121 LearnersLast updated on May 26th, 2025
Square Root of 404
If a number is multiplied by itself, the result is a square. The inverse operation is finding the square root. Square roots are used in various fields such as engineering, physics, and finance. Here, we will discuss the square root of 404.
What is the Square Root of 404?
The square root is the inverse operation of squaring a number. 404 is not a perfect square. The square root of 404 can be expressed in both radical and exponential form. In radical form, it is expressed as √404, whereas (404)^(1/2) is the exponential form. √404 ≈ 20.099, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
Finding the Square Root of 404
The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, the long division method and approximation method are used. Let's learn these methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 404 by Prime Factorization Method
The product of prime factors represents the prime factorization of a number. Let's see how 404 is broken down into its prime factors:
Step 1: Finding the prime factors of 404
Breaking it down, we get 2 x 2 x 101: 2² x 101
Step 2: Since 404 is not a perfect square, its prime factors cannot be grouped into pairs. Therefore, calculating √404 using prime factorization directly is not feasible.
Square Root of 404 by Long Division Method
The long division method is useful for non-perfect square numbers. We need to find the closest perfect square number to 404. Here's how to find the square root using the long division method, step by step:
Step 1: Group the digits of 404 from right to left. In this case, we group it as 04 and 4.
Step 2: Find n, whose square is less than or equal to 4. Here, n is 2, since 2² = 4.
Step 3: Subtract 4 from 4, the remainder is 0. Bring down 04 to make the new dividend 4.
Step 4: Double the divisor (2) and write it as 4.
Step 5: Find a digit, n, such that 4n × n ≤ 400. Here, n is 0, since 40 × 0 = 0, which is ≤ 4.
Step 6: Subtract 0 from 4, the remainder is 4.
Step 7: Add a decimal point and bring down two zeros to make the new dividend 400.
Step 8: Now, find a new digit, n, such that 40n × n ≤ 400. Here, n is 9, since 409 × 9 = 3681, which is ≤ 4000.
Step 9: Continue these steps until you achieve two decimal places.
The approximate square root of 404 is 20.099.
Square Root of 404 by Approximation Method
The approximation method is another way to find square roots, especially for non-perfect squares. Here's how to approximate the square root of 404:
Step 1: Find the closest perfect squares around 404.
The closest perfect squares are 400 (20²) and 441 (21²). Therefore, √404 is between 20 and 21.
Step 2: Use linear approximation: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square) (404 - 400) / (441 - 400) = 4 / 41 ≈ 0.098.
Add this to the lower square root: 20 + 0.098 = 20.098.
Thus, √404 ≈ 20.098.
Common Mistakes and How to Avoid Them in the Square Root of 404
Students may make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in the long division method. Let's explore a few common mistakes and how to avoid them.
Mistake 1
Forgetting about the negative square root
It's important to remember that every positive number has both positive and negative square roots. Generally, we focus on the positive square root.
For example, while √404 ≈ 20.099, there is also -20.099, which should not be overlooked.
Square Root of 404 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √404?
The area of the square is 404 square units.
Explanation
The area of a square = side².
The side length is given as √404. Area of the square = (√404)² = 404.
Therefore, the area of the square box is 404 square units.
Problem 2
A square-shaped building measuring 404 square feet is built; if each side is √404, what will be the square feet of half of the building?
202 square feet.
Explanation
Since the building is square-shaped, divide the total area by 2.
404 / 2 = 202. So, half of the building measures 202 square feet.
Problem 3
Calculate √404 × 5.
100.495
Explanation
First, find the square root of 404, which is approximately 20.099.
Then, multiply 20.099 by 5: 20.099 × 5 ≈ 100.495.
Problem 4
What will be the square root of (200 + 204)?
The square root is 20.
Explanation
First, find the sum of (200 + 204): 200 + 204 = 404.
The square root of 404 is approximately ±20.099.
Therefore, the square root of (200 + 204) is ±20.099.
Problem 5
Find the perimeter of a rectangle if its length ‘l’ is √404 units and the width ‘w’ is 50 units.
The perimeter of the rectangle is approximately 140.198 units.
Explanation
Perimeter of a rectangle = 2 × (length + width).
Perimeter = 2 × (√404 + 50) ≈ 2 × (20.099 + 50) = 2 × 70.099 ≈ 140.
198 units.
FAQ on Square Root of 404
1.What is √404 in its simplest form?
2.Mention the factors of 404.
3.Calculate the square of 404.
4.Is 404 a prime number?
5.404 is divisible by?
Important Glossaries for the Square Root of 404
- Square root: The square root is the operation that finds a number which, when multiplied by itself, yields the original number. Example: √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.
- Principal square root: The non-negative square root of a number. For example, the principal square root of 64 is 8.
- Perfect square: A number that is the square of an integer. For example, 36 is a perfect square because it is 6².
- Decimal: A number that includes a whole number and a fractional part separated by a decimal point, such as 7.86.
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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.
