Square Root of 648

The square root is the inverse of the square of the number. 648 is not a perfect square. The square root of 648 is expressed in both radical and exponential form. In the radical form, it is expressed as √648, whereas (648)^(1/2) in the exponential form. √648 ≈ 25.45584, 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.

For perfect square numbers, the prime factorization method is used. 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

The product of prime factors is the prime factorization of a number. Now let us look at how 648 is broken down into its prime factors.

Step 1: Finding the prime factors of 648 Breaking it down, we get 2 x 2 x 2 x 3 x 3 x 3 x 3: 2^3 x 3^4

Step 2: Now we have found the prime factors of 648. The second step is to make pairs of those prime factors. Since 648 is not a perfect square, the digits of the number can’t be grouped in perfect pairs.

Therefore, calculating 648 using prime factorization alone for an exact square root is impractical.

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 numbers from right to left. In the case of 648, we group it as 48 and 6.

Step 2: Now find n whose square is less than or equal to 6. We can say n is 2 because 2 x 2 = 4, which is less than or equal to 6. Now, subtract 4 from 6 and get a remainder of 2.

Step 3: Bring down 48, making the new dividend 248. Add the old divisor (2) to itself, resulting in a new divisor of 4.

Step 4: Now find a number m such that 4m x m is less than or equal to 248. We find m as 6, because 46 x 6 = 276, which exceeds 248, so try m = 5, then 45 x 5 = 225.

Step 5: Subtract 225 from 248, resulting in 23. The quotient is now 25.

Step 6: Since the dividend is less than the divisor, add a decimal point and bring down two zeroes, getting 2300 as the new dividend.

Step 7: Find the new divisor that is 50 because 505 x 5 = 2525. Instead, try 504 x 4 = 2016.

Step 8: Subtract 2016 from 2300, getting 284.

Step 9: The quotient is now 25.4. Continue this process until you get two numbers after the decimal point or until the remainder is zero.

So, the square root of √648 ≈ 25.45.

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 648 using the approximation method.

Step 1: Find the closest perfect squares to √648. The smallest perfect square less than 648 is 625, and the largest perfect square greater than 648 is 676. √648 falls somewhere between 25 and 26.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (648 - 625) / (676 - 625) = 23/51 ≈ 0.451

The approximate square root is 25 + 0.451 = 25.451, so the square root of 648 is approximately 25.451.

• 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.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because it is 6^2.
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors. For example, the prime factorization of 648 is 2^3 x 3^4.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing the number into parts, using a divisor, and finding the quotient step by step.