Square Root of 11336

The square root is the inverse operation of squaring a number. 11336 is a perfect square. The square root of 11336 is expressed in both radical and exponential forms. In the radical form, it is expressed as √11336, whereas in the exponential form, it is expressed as (11336)^(1/2). √11336 = 106, 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 like 11336. Other methods such as the long division method and approximation can also be used based on the requirement. 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 11336 is broken down into its prime factors.

Step 1: Finding the prime factors of 11336

Breaking it down, we get 2 x 2 x 2 x 7 x 7 x 29 x 29: 2^3 x 7^2 x 29^2

Step 2: Now we have found the prime factors of 11336. We make pairs of those prime factors. Since 11336 is a perfect square, the digits of the number can be grouped in pairs. Therefore, calculating √11336 using prime factorization is possible. Step 3: Pair the prime factors: (2 x 7 x 29) x (2 x 7 x 29) = (406)^2. Therefore, the square root of 11336 is 106.

The long division method is used for finding the square root of both perfect and non-perfect square numbers. Let us now learn how to find the square root using the long division method, step by step.

Step 1: Start by grouping the digits into pairs from right to left. In the case of 11336, we group it as 11 and 336.

Step 2: Find a number whose square is less than or equal to 11. Here, 3^2 = 9 is the closest. Put 3 as the first digit of the quotient.

Step 3: Subtract 9 from 11, giving a remainder of 2. Bring down the next pair, 336, making the new dividend 236.

Step 4: Double the quotient obtained so far (which is 3), giving 6, and use it as the new divisor's first digit. Find a digit "n" such that 6n x n ≤ 236.

Step 5: We find 66 x 6 = 396, which is too large, so we try 64 x 4 = 256. But for 63 x 3 = 189, which is less than or equal to 236.

Step 6: Subtract 189 from 236. The remainder is 47.

Step 7: Bring down two zeros to make the new dividend 4700.

Step 8: Double the current quotient (33) and use it to form a new divisor. Find a digit "n" such that 66n x n is less than or equal to 4700.

Step 9: Continue this process until you have enough decimal places or the remainder becomes zero.

The square root of 11336 is 106.

The approximation method is another method for finding square roots. It is a simple way to estimate the square root of a given number. Let us learn how to approximate the square root of 11336.

Step 1: Find the closest perfect squares around 11336. The closest perfect squares are 10000 (100^2) and 12100 (110^2). Therefore, √11336 falls somewhere between 100 and 110.

Step 2: Use the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula: (11336 - 10000) / (12100 - 10000) = 1336 / 2100 ≈ 0.636

Step 3: Add this decimal to the lower square root value: 100 + 0.636 = 100.636, but since 11336 is a perfect square, we know √11336 is exactly 106.

• Square root: A square root is the inverse of squaring a number. For example, 10^2 = 100, and the inverse of the square is the square root, √100 = 10.
   
• Rational number: A rational number can be expressed in the form of p/q, where p and q are integers and q ≠ 0.
   
• Principal square root: A number has both positive and negative square roots; the principal square root is the non-negative one used in most contexts.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 144 is a perfect square because it is 12^2.
   
• Prime factorization: Expressing a number as a product of its prime factors. For example, the prime factorization of 28 is 2^2 x 7.