Square Root of 1824
2026-02-28 00:46 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 fields such as vehicle design, finance, and more. Here, we will discuss the square root of 1824.

What is the Square Root of 1824?

The square root is the inverse of the square of the number. 1824 is not a perfect square. The square root of 1824 is expressed in both radical and exponential forms. In radical form, it is expressed as √1824, whereas in exponential form it is expressed as (1824)^(1/2). √1824 ≈ 42.708, which is an irrational number because it cannot be expressed as a fraction of two integers.

Finding the Square Root of 1824

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers, where 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 1824 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 1824 is broken down into its prime factors:

Step 1: Finding the prime factors of 1824 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 3 x 19 = 2^5 x 3 x 19

Step 2: Now we found the prime factors of 1824. The second step is to make pairs of those prime factors. Since 1824 is not a perfect square, the digits of the number can’t be grouped in pairs evenly.

Therefore, calculating 1824 using prime factorization directly is complex, and other methods are preferred.

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Square Root of 1824 by Long Division Method

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step:

Step 1: To begin with, group the digits in pairs from right to left. In the case of 1824, group them as 18 and 24.

Step 2: Find n whose square is less than or equal to 18. We can say n is 4 because 4^2 = 16, which is less than 18. Now the quotient is 4, and the remainder is 2.

Step 3: Bring down the next pair 24, making the new dividend 224. Add the old divisor with the same number: 4 + 4 = 8, which will be our new divisor.

Step 4: The new divisor will be 8n. We need to find the value of n such that 8n x n ≤ 224.

Step 5: Consider n as 2. Then 82 x 2 = 164.

Step 6: Subtract 164 from 224, the difference is 60, and the quotient is 42.

Step 7: Since the dividend is less than the divisor, add a decimal point and bring down two zeros, making the dividend 6000.

Step 8: Find n such that 842 x n ≤ 6000. Let n be 7. Then 842 x 7 = 5894.

Step 9: Subtract 5894 from 6000, resulting in 106.

Step 10: The quotient is 42.7.

Step 11: Continue these steps until you get two decimal places.

So the square root of √1824 is approximately 42.708.

Square Root of 1824 by Approximation Method

Approximation is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 1824 using the approximation method.

Step 1: Find the closest perfect squares around √1824. The smallest perfect square less than 1824 is 1764 (42^2), and the largest perfect square greater than 1824 is 1849 (43^2). √1824 falls between 42 and 43.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). (1824 - 1764) / (1849 - 1764) = 60/85 ≈ 0.70588

Using the formula, add the result to the lower perfect square root: 42 + 0.70588 ≈ 42.71, so the square root of 1824 is approximately 42.71.

Common Mistakes and How to Avoid Them in the Square Root of 1824

Students often make mistakes while finding the square root, such as forgetting about the negative square root or skipping long division steps. Let's look at a few 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 √1824?

Okay, lets begin

The area of the square is approximately 3324.096 square units.

Explanation

The area of the square = side^2.

The side length is given as √1824.

Area of the square = side^2 = √1824 x √1824 ≈ 42.708 x 42.708 ≈ 1824.

Therefore, the area of the square box is approximately 3324.096 square units.

Well explained 👍

Problem 2

A square-shaped garden measuring 1824 square feet is built; if each of the sides is √1824, what will be the square feet of half of the garden?

Okay, lets begin

912 square feet

Explanation

Since the garden is square-shaped, we can divide the total area by 2 to find half of it.

Dividing 1824 by 2 gives 912. So half of the garden measures 912 square feet.

Well explained 👍

Problem 3

Calculate √1824 x 5.

Okay, lets begin

Approximately 213.54

Explanation

First, find the square root of 1824, which is approximately 42.708.

Then multiply 42.708 by 5. So, 42.708 x 5 ≈ 213.54.

Well explained 👍

Problem 4

What will be the square root of (1624 + 200)?

Okay, lets begin

The square root is approximately 43.

Explanation

To find the square root, first find the sum of (1624 + 200).

1624 + 200 = 1824.

The square root of 1824 is approximately 42.708.

Therefore, the square root of (1624 + 200) is approximately 42.708.

Well explained 👍

Problem 5

Find the perimeter of a rectangle if its length ‘l’ is √1824 units and the width ‘w’ is 38 units.

Okay, lets begin

The perimeter of the rectangle is approximately 161.42 units.

Explanation

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

Perimeter = 2 × (√1824 + 38) ≈ 2 × (42.708 + 38) ≈ 2 × 80.708 ≈ 161.42 units.

Well explained 👍

FAQ on Square Root of 1824

1.What is √1824 in its simplest form?

The prime factorization of 1824 is 2^5 x 3 x 19, so the simplest form of √1824 cannot be further simplified exactly, as it is an irrational number.

2.Mention the factors of 1824.

Factors of 1824 include 1, 2, 3, 4, 6, 8, 12, 16, 19, 24, 38, 48, 57, 76, 114, 152, 228, 304, 456, 608, 912, and 1824.

3.Calculate the square of 1824.

We get the square of 1824 by multiplying the number by itself: 1824 x 1824 = 3,328,576.

4.Is 1824 a prime number?

1824 is not a prime number, as it has more than two factors.

5.1824 is divisible by?

1824 has many factors; it is divisible by 1, 2, 3, 4, 6, 8, 12, 16, 19, 24, 38, 48, 57, 76, 114, 152, 228, 304, 456, 608, 912, and 1824.

Important Glossaries for the Square Root of 1824

  • Square root: A square root is the inverse operation of squaring a number. For example, if 5^2 = 25, then the square root of 25 is √25 = 5.
  • Irrational number: An irrational number cannot be written as a simple fraction; its decimal form is non-repeating and non-terminating. For example, √2 is irrational.
  • Prime factorization: Expressing a number as the product of its prime factors. For example, the prime factorization of 1824 is 2^5 x 3 x 19.
  • Long division method: A method used to find the square root of a number that is not a perfect square by dividing the number into pairs of digits and using a series of steps.
  • Approximation method: A method used to estimate the square root of a number by identifying nearby perfect squares and using interpolation to find a more precise value.

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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.

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: He loves to play the quiz with kids through algebra to make kids love it.