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.

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

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.

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.

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.

• 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.