Square Root of 11536

The square root is the inverse of the square of the number. 11536 is a perfect square. The square root of 11536 can be expressed in both radical and exponential form. In the radical form, it is expressed as √11536, whereas in the exponential form, it is expressed as (11536)^(1/2). √11536 = 107, 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.

For perfect square numbers, the prime factorization method is used. The long division method can also be used to find square roots, especially for non-perfect squares. 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 11536 is broken down into its prime factors.

Step 1: Finding the prime factors of 11536.

Breaking it down, we get 2 × 2 × 2 × 2 × 29 × 29: 2^4 × 29^2

Step 2: Now that we have found the prime factors of 11536, the next step is to make pairs of those prime factors. Since 11536 is a perfect square, we can group the digits in pairs. Therefore, calculating √11536 using prime factorization is possible.

The long division method is particularly useful for finding square roots of non-perfect square numbers, but it can also confirm perfect squares. Let us learn how to find the square root using the long division method, step by step.

Step 1: Begin by grouping the numbers from right to left. In the case of 11536, we group it as 11 and 536.

Step 2: Find the largest number whose square is less than or equal to 11. The number is 3, because 3 × 3 = 9. The quotient is 3, and after subtracting, the remainder is 2.

Step 3: Bring down 536 to make the new dividend 2536. Double the quotient (3) to get the new divisor's first digit, 6.

Step 4: Find a digit to complete the divisor such that 6x × x is less than or equal to 2536, where x is the digit. Here, x is 7, because 67 × 7 = 469, and 469 < 2536.

Step 5: Subtract 469 from 2536 to get the remainder 2067.

Step 6: Bring down zeros and continue the process until the remainder is zero or to the desired decimal precision.

So the square root of 11536 is 107.

The approximation method is less useful for perfect squares but can provide a quick estimation for non-perfect squares. Let us see how to find the square root of 11536 using this method.

Step 1: Identify the closest perfect squares to 11536. In this case, 11536 itself is a perfect square.

Step 2: Since 11536 is a perfect square, its square root is exactly 107.

Thus, the approximation confirms that the square root of 11536 is 107.

• 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, which is √100 = 10.
   
• Rational number: A rational number is a number that can be expressed 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: 36 is a perfect square because it is 6 × 6.
   
• Prime factorization: Expressing a number as the product of its prime factors. Example: The prime factorization of 72 is 2^3 × 3^2.
   
• Long division method: A method used for finding the square root of a number by dividing it into parts, useful for non-perfect squares and perfect squares alike.