Square Root of 11025

The square root is the inverse of the square of the number. 11025 is a perfect square. The square root of 11025 is expressed in both radical and exponential form. In the radical form, it is expressed as √11025, whereas (11025)^(1/2) in the exponential form. √11025 = 105, which is a rational number because it can 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. For perfect squares like 11025, the prime factorization method is very effective. 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 11025 is broken down into its prime factors.

Step 1: Finding the prime factors of 11025

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

Step 2: Now we found out the prime factors of 11025. The second step is to make pairs of those prime factors. Since 11025 is a perfect square, we can make pairs of each prime factor: (3 x 3), (5 x 5), and (7 x 7). The square root of 11025 is the product of one factor from each pair, which is 3 x 5 x 7 = 105.

The long division method can also be used for perfect squares. 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 11025, we need to group it as 25 and 110.

Step 2: Now we need to find n whose square is less than or equal to 11. We can say n as ‘3’ because 3 x 3 = 9 is lesser than or equal to 11. Now the quotient is 3, and after subtracting 9 from 11, the remainder is 2.

Step 3: Now let us bring down 02, making the new dividend 202. Add the old divisor with the same number 3 + 3 to get 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we have 6n as the new divisor, and we need to find the value of n such that 60n x n ≤ 202

Step 5: Considering n as 3, we have 63 x 3 = 189.

Step 6: Subtract 189 from 202, the difference is 13, and the quotient is 33.

Step 7: Bring down 25, making the new dividend 1325. Add a decimal point to the quotient.

Step 8: The new divisor will be 660. Finding n such that 660n x n ≤ 1325, we get n = 2.

Step 9: The remainder becomes 0 after subtracting, and the quotient is 105.

The approximation method is another method for finding the square roots, but for a perfect square like 11025, the exact method is more appropriate. Now let us learn how to find the square root of 11025 using approximation.

Step 1: Now we have to find the closest perfect square of √11025. The closest perfect squares around 11025 are 10816 (104^2) and 11236 (106^2). √11025 falls exactly at 105, which is between 104 and 106.

Step 2: The square root of 11025 is precisely 105.

• Square root: A square root is the inverse of a square. Example: 10^2 = 100 and the inverse of the square is the square root, that is, √100 = 10.
   
• Rational number: A rational number is a number that can be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Perfect square: A perfect square is a number that is the square of an integer. Example: 144 is a perfect square since it is 12^2.
   
• Prime factorization: Prime factorization is the process of expressing a number as a product of its prime factors. For example, the prime factorization of 28 is 2 x 2 x 7.
   
• Long division method: A method used to find the square root of a number by grouping digits, finding divisors, and calculating remainders systematically.