Square Root of 4525

The square root is the inverse of the square of the number. 4525 is not a perfect square. The square root of 4525 is expressed in both radical and exponential form. In the radical form, it is expressed as √4525, whereas (4525)^(1/2) in the exponential form. √4525 ≈ 67.2959, 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 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 4525 is broken down into its prime factors:

Step 1: Finding the prime factors of 4525

Breaking it down, we get 5 x 5 x 181: 5^2 x 181

Step 2: Now we found out the prime factors of 4525. The second step is to make pairs of those prime factors. Since 4525 is not a perfect square, therefore the digits of the number can’t be grouped in pairs completely. Therefore, calculating 4525 using prime factorization is not straightforward.

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 4525, we need to group it as 45 and 25.

Step 2: Now we need to find n whose square is less than or equal to 45. We can say n as ‘6’ because 6 x 6 = 36, which is less than 45. Now the quotient is 6, after subtracting 36 from 45, the remainder is 9.

Step 3: Now let us bring down 25, which is the new dividend. Add the old divisor with the same number, 6 + 6, we 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, we need to find the value of n.

Step 5: The next step is finding 12n × n ≤ 925. Let us consider n as 7, now 127 x 7 = 889.

Step 6: Subtract 925 from 889, the difference is 36, and the quotient is 67.

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 that is 134, because 1344 x 4 = 5376, which is greater than 3600. Therefore, 1343 x 3 = 4029, which is too much, so we try 1342 x 2 = 2684.

Step 9: Subtracting 2684 from 3600, we get the result 916.

Step 10: Now the quotient is 67.2.

Step 11: Continue doing these steps until we get two numbers after the decimal point or until the remainder is zero.

So the square root of √4525 ≈ 67.29.

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 4525 using the approximation method.

Step 1: Now we have to find the closest perfect square of √4525. The smallest perfect square less than 4525 is 4356 (66^2) and the largest perfect square greater than 4525 is 4624 (68^2). √4525 falls somewhere between 66 and 68.

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 (4525 - 4356) ÷ (4624 - 4356) ≈ 0.2959. 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 66 + 0.2959 ≈ 66.2959, so the square root of 4525 is approximately 67.30.

• 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 is more prominent 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, 64 is a perfect square because it is 8^2.

• 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.