Square Root of 504

The square root is the inverse of the square of the number. 504 is not a perfect square. The square root of 504 is expressed in both radical and exponential form. In the radical form, it is expressed as √504, whereas (504)^(1/2) in the exponential form. √504 ≈ 22.44994, 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 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 504 is broken down into its prime factors.

Step 1: Finding the prime factors of 504

Breaking it down, we get 2 × 2 × 2 × 3 × 3 × 7: 2^3 × 3^2 × 7

Step 2: Now we found out the prime factors of 504. The second step is to make pairs of those prime factors. Since 504 is not a perfect square, therefore the digits of the number can’t be grouped into pairs. Therefore, calculating 504 using prime factorization is impossible.

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 504, we need to group it as 04 and 50.

Step 2: Now we need to find n whose square is closest to or less than 50. We can say n is '7' because 7 × 7 = 49, which is less than 50. After subtracting 50 - 49, the remainder is 1.

Step 3: Bring down 04 to get 104 as the new dividend. Add the old divisor with the same number 7 + 7 to get 14, which will be our new divisor.

Step 4: The new divisor is 14n. We need to find the value of n where 14n × n ≤ 104. Let us consider n as 7, now 147 × 7 = 1029.

Step 5: Subtract 1040 from 1029, the difference is 11. The quotient is 22.4 so far.

Step 6: 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 1100.

Step 7: Now we need to find the new divisor that is 22 because 224 × 22 = 11000.

Step 8: Subtracting 11000 from 1100 gives us a result of 0.

Step 9: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.

So the square root of √504 is approximately 22.45.

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 504 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √504. The smallest perfect square less than 504 is 484, and the largest perfect square more than 504 is 529. √504 falls somewhere between 22 and 23.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (504 - 484) ÷ (529 - 484) = 20 ÷ 45 ≈ 0.444

Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 22 + 0.444 = 22.444.

• 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: The expression of a number as the product of its prime factors. Example: The prime factorization of 504 is 2^3 × 3^2 × 7.
   
• Long division method: A step-by-step process to find the square root of a number by dividing and finding the quotient, used especially for non-perfect squares.