Square Root of 524

The square root is the inverse of the square of the number. 524 is not a perfect square. The square root of 524 is expressed in both radical and exponential form. In the radical form, it is expressed as √524, whereas (524)^(1/2) in the exponential form. √524 ≈ 22.876, 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:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 524 is broken down into its prime factors.

Step 1: Finding the prime factors of 524 Breaking it down, we get 2 x 2 x 131: 2² x 131¹

Step 2: Now we found out the prime factors of 524. The second step is to make pairs of those prime factors. Since 524 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 524 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 524, we need to group it as 24 and 5.

Step 2: Now we need to find n whose square is 5. We can say n as ‘2’ because 2 x 2 is lesser than or equal to 5. Now the quotient is 2; after subtracting 4 (2 x 2) from 5, the remainder is 1.

Step 3: Now let us bring down 24, which is the new dividend. Add the old divisor with the same number: 2 + 2, we get 4, which will be our new divisor.

Step 4: The new divisor will be the sum of the current divisor and quotient. Now we get 4n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 4n x n ≤ 124. Let us consider n as 2; now 42 x 2 = 84.

Step 6: Subtract 124 from 84; the difference is 40, and the quotient is 22.

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 4000.

Step 8: Now we need to find the new divisor that is 229 because 229 x 9 = 2061.

Step 9: Subtracting 2061 from 4000, we get the result 1939.

Step 10: Now the quotient is 22.8.

Step 11: 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 √524 is approximately 22.87.

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 524 using the approximation method.

Step 1: Now we have to find the closest perfect square of √524. The smallest perfect square of 524 is 484, and the largest perfect square of 524 is 529. √524 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 (524 - 484) / (529 - 484) = 0.888.

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 + 0.888 ≈ 22.888, so the square root of 524 is approximately 22.888.

• Square root: A square root is the inverse of a square. Example: 4² = 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 a 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.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 16 is a perfect square because it equals 4².

• Long division method: A mathematical technique used to find the square root of a non-perfect square by dividing the number into groups and solving step by step.