Square Root of 9225

The square root is the inverse of the square of a number. 9225 is not a perfect square. The square root of 9225 is expressed in both radical and exponential form. In the radical form, it is expressed as √9225, whereas (9225)^(1/2) in the exponential form. √9225 ≈ 96.06559, 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 9225 is broken down into its prime factors:

Step 1: Finding the prime factors of 9225

Breaking it down, we get 3 x 3 x 5 x 5 x 41 = 3² x 5² x 41

Step 2: Now that we have found the prime factors of 9225, we make pairs of those prime factors. Since 9225 is not a perfect square, the digits of the number can’t be grouped into complete pairs. Therefore, calculating 9225 using prime factorization alone is insufficient.

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 9225, we need to group it as 25 and 92.

Step 2: Now we need to find n whose square is less than or equal to 92. We can say n is ‘9’ because 9 x 9 = 81, which is less than 92. Now the quotient is 9, and after subtracting 81 from 92, the remainder is 11.

Step 3: Now, let us bring down 25, which becomes the new dividend. Add the old divisor with the same number 9 + 9 to get 18, which will be our new divisor.

Step 4: The new divisor will be represented as 18n. We need to find the value of n such that 18n x n ≤ 1125.

Step 5: The next step is finding 18n × n ≤ 1125. Let us consider n as 6, now 186 x 6 = 1116.

Step 6: Subtract 1116 from 1125; the difference is 9, and the quotient becomes 96.

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 900.

Step 8: Now we need to find the new divisor, which is 193, because 193 x 4 = 772.

Step 9: Subtracting 772 from 900, we get the result 128.

Step 10: Now the quotient is 96.0.

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

So the square root of √9225 is approximately 96.06.

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 9225 using the approximation method.

Step 1: We have to find the closest perfect squares to √9225. The smallest perfect square less than 9225 is 9216 (96²), and the largest perfect square greater than 9225 is 9409 (97²). Thus, √9225 falls between 96 and 97.

Step 2: Now we need to apply the approximation formula: (Given number - smaller perfect square) ÷ (larger perfect square - smaller perfect square) Applying the formula: (9225 - 9216) ÷ (9409 - 9216) = 9 ÷ 193 ≈ 0.0466

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 96 + 0.0466 ≈ 96.0466, so the square root of 9225 is approximately 96.05.

• 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 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 process of expressing a number as a product of its prime factors.
   
• Long division method: A technique for finding the square root of a number through repeated division and approximation steps.