Square Root of 9100

The square root is the inverse of the square of the number. 9100 is not a perfect square. The square root of 9100 is expressed in both radical and exponential form. In the radical form, it is expressed as √9100, whereas (9100)^(1/2) in the exponential form. √9100 ≈ 95.3948, 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 9100 is broken down into its prime factors.

Step 1: Finding the prime factors of 9100

Breaking it down, we get 2 x 2 x 5 x 5 x 7 x 13: 2^2 x 5^2 x 7 x 13

Step 2: Now we found out the prime factors of 9100. The second step is to make pairs of those prime factors. Since 9100 is not a perfect square, therefore, calculating 9100 using prime factorization is more complex without perfect pairs.

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 9100, we need to group it as 00 and 91.

Step 2: Now we need to find n whose square is less than or equal to 91. We consider n as '9' because 9 x 9 = 81 is less than 91. Now the quotient is 9, and after subtracting 81 from 91, the remainder is 10.

Step 3: Now let us bring down 00 to make the new dividend 1000. Add the old divisor, 9, with the same number 9 + 9, we get 18, which will be our new divisor.

Step 4: The new divisor will be 18n, where we need to find the value of n such that 18n x n is less than or equal to 1000.

Step 5: The next step is finding 18n x n ≤ 1000. Let us consider n as 5, then 185 x 5 = 925.

Step 6: Subtract 925 from 1000, the difference is 75, and the quotient is 95.

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 7500.

Step 8: Now we need to find the new divisor, which is 953, because 953 x 8 = 7624.

Step 9: Subtracting 7624 from 7500 gives us a negative remainder, so we adjust n to 7 and get 952 x 7 = 6664.

Step 10: Subtracting 6664 from 7500, we get the remainder 836.

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

So, the square root of √9100 is approximately 95.39.

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 9100 using the approximation method.

Step 1: Now we have to find the closest perfect square of √9100. The smallest perfect square less than 9100 is 9025, and the largest perfect square greater than 9100 is 9216. √9100 falls somewhere between 95 and 96.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula, (9100 - 9025) ÷ (9216 - 9025) = 0.3948.

Using the formula, we identified the decimal point of our square root. The next step is adding the integer part with the decimal number, which is 95 + 0.3948 ≈ 95.39, so the square root of 9100 is approximately 95.39.

• 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.
   
• Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors. For example, the prime factorization of 9100 is 2 x 2 x 5 x 5 x 7 x 13.
   
• Long division method: A method used to find the square root of a number, especially for non-perfect squares, involving iterative division and subtraction.
   
• Approximation method: A method used to estimate the square root of a number by finding the closest perfect squares and using a formula to approximate the value.