Square Root of 4725

The square root is the inverse of the square of the number. 4725 is not a perfect square. The square root of 4725 is expressed in both radical and exponential form. In the radical form, it is expressed as √4725, whereas (4725)^(1/2) in exponential form. √4725 ≈ 68.726, 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 4725 is broken down into its prime factors.

Step 1: Finding the prime factors of 4725

Breaking it down, we get 3 x 3 x 5 x 5 x 7 x 9: 3^2 x 5^2 x 7 x 3

Step 2: Now we found the prime factors of 4725. The second step is to make pairs of those prime factors. Since 4725 is not a perfect square, the digits of the number can’t be grouped in pairs evenly. Therefore, calculating 4725 using prime factorization directly to obtain a square root is not feasible.

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 4725, we need to group it as 25 and 47.

Step 2: Now we need to find n whose square is closest to 47. We can say n is ‘6’ because 6 x 6 = 36, which is less than 47. The quotient is 6, and after subtracting 36 from 47, the remainder is 11.

Step 3: Now let us bring down 25, which is the new dividend. Add the old divisor with the same number, 6 + 6, to get 12, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 12n as the new divisor, where n needs to be found.

Step 5: The next step is finding 12n x n ≤ 1125. Let us consider n as 9, now 12 x 9 x 9 = 1089.

Step 6: Subtract 1125 from 1089; the difference is 36, and the quotient is 69.

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 3600.

Step 8: Now we need to find the new divisor, which is 138 because 138 x 2 = 276.

Step 9: Subtracting 276 from 3600, we get the result 324.

Step 10: Now the quotient is 68.72.

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

So the square root of √4725 ≈ 68.726.

The approximation method is another method for finding the 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 4725 using the approximation method.

Step 1: Now we have to find the closest perfect square of √4725. The smallest perfect square less than 4725 is 4624 (68^2) and the largest perfect square is 4761 (69^2). √4725 falls somewhere between 68 and 69.

Step 2: Now we need to apply the formula that is: (Given number - smallest perfect square) / (Larger perfect square - smallest perfect square) Going by the formula (4725 - 4624) ÷ (4761 - 4624) = 101 ÷ 137 ≈ 0.737 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 68 + 0.737 ≈ 68.737, so the square root of 4725 is approximately 68.737.

• 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 a principal square root.

• Prime factorization: This method involves breaking down a number into its prime factors to simplify calculations, especially useful for finding square roots of perfect squares.

• Long division method: A systematic method used to find the square root of a non-perfect square by dividing the number step by step to approximate its square root.