Square Root of 6144

The square root is the inverse of the square of the number. 6144 is not a perfect square. The square root of 6144 is expressed in both radical and exponential form. In the radical form, it is expressed as √6144, whereas (6144)^(1/2) in the exponential form. √6144 ≈ 78.4288, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where long-division and approximation methods 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 6144 is broken down into its prime factors.

Step 1: Finding the prime factors of 6144 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 4: 2^9 x 3 x 4

Step 2: Now we found out the prime factors of 6144. The next step is to make pairs of those prime factors. Since 6144 is not a perfect square, therefore the digits of the number can’t be grouped in perfect pairs.

Therefore, calculating 6144 using prime factorization in this context will not yield an integer result for its square root.

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, we need to group the numbers from right to left. In the case of 6144, we need to group it as 44 and 61.

Step 2: Now we need to find n whose square is less than or equal to 61. We can say n as ‘7’ because 7^2 = 49 is lesser than 61. Now the quotient is 7, and after subtracting 49 from 61, the remainder is 12.

Step 3: Now let us bring down 44, which is the new dividend. Add the old divisor with the same number, 7 + 7, we get 14, which will be our new divisor.

Step 4: The new divisor will be 14n, where n is to be found such that 14n x n is less than or equal to 1244.

Step 5: We find n = 8 as 148 x 8 = 1184 is less than 1244

Step 6: Subtract 1184 from 1244, the difference is 60.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 6000.

Step 8: Now we need to find the new divisor, which would be 1578, since 1578 x 3 = 4734

Step 9: Subtracting 4734 from 6000, we get the result 1266.

Step 10: The quotient is approximately 78.4.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal values continue till the remainder is zero

So the square root of √6144 ≈ 78.4288

The approximation method 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 6144 using the approximation method.

Step 1: Find the closest perfect square of √6144.

The smallest perfect square less than 6144 is 6084 (78^2) and the largest perfect square greater than 6144 is 6241 (79^2). √6144 falls somewhere between 78 and 79.

Step 2: Now apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square) Using the formula: (6144 - 6084) ÷ (6241 - 6084) = 60 ÷ 157 ≈ 0.382 Adding the decimal to the whole number, we get 78 + 0.382 = 78.382.

Thus, the square root of 6144 is approximately 78.382.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16 and the inverse of the square is the square root, that is, √16 = 4.

• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero, and p and q are integers.

• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.

• Prime factorization: Prime factorization is expressing a number as the product of its prime numbers. For example, the prime factorization of 18 is 2 x 3^2.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.