Square Root of 11849

The square root is the inverse of squaring a number. 11849 is a perfect square. The square root of 11849 can be expressed in both radical and exponential form. In the radical form, it is expressed as √11849, whereas in exponential form it is expressed as (11849)^(1/2). √11849 = 109, 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 non-perfect squares, methods like the long division method and approximation method are used. Let us now learn the following methods:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 11849 is broken down into its prime factors.

Step 1: Finding the prime factors of 11849 Breaking it down, we get 109 × 109.

Step 2: Now we found out the prime factors of 11849. Since 11849 is a perfect square, the digits of the number can be grouped in pairs.

Therefore, the square root of 11849 using prime factorization is 109.

The long division method is particularly used for both perfect and non-perfect square numbers. Let's 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 11849, we group it as 11, 84, and 9.

Step 2: Find a number whose square is less than or equal to 11. We can select 3, as 3 × 3 = 9 is less than 11. The initial quotient is 3, and after subtracting 9 from 11, the remainder is 2.

Step 3: Bring down the next pair of digits (84) to get a new dividend of 284. Double the quotient (3) to get a new divisor base of 6.

Step 4: Select a digit (4) such that 64 × 4 ≤ 284. 64 × 4 = 256, which is less than 284. Subtract 256 from 284 to get a remainder of 28.

Step 5: Bring down the next pair of digits (9) to get a new dividend of 289.

Step 6: Double the part of the quotient found so far (34) to get 68 and find a digit (1) such that 681 × 1 = 681 is less than or equal to 289. Subtract 681 from 289 to get a remainder of zero.

Step 7: As the remainder is zero, the square root of 11849 is 109.

The approximation method is another way to find square roots. It is an easy method to find the square root of a given number. Let's learn how to find the square root of 11849 using the approximation method.

Step 1: Identify two perfect squares between which 11849 lies. The smaller perfect square is 10404 (102^2), and the larger perfect square is 11664 (108²). √11849 falls between 108 and 109.

Step 2: Use interpolation to approximate the square root: (11849 - 11664) / (11881 - 11664) = 0.5 Adding this to 108, we get 108 + 0.5 = 108.5, which is an approximation. The exact value, however, is 109.

• Square root: A square root is the inverse of squaring a number. Example: 4² = 16 and the inverse of the square is the square root, that is √16 = 4.

• Perfect square: A perfect square is an integer that is the square of an integer. For example, 16 is a perfect square because it is 4 squared.

• Rational number: A rational number is a number that can be expressed as the quotient or fraction p/q, where p and q are integers and q is not zero.

• Integer: Integers are a set of numbers that include all whole numbers and their negatives. For example, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, etc., are integers.

• Approximation: Approximation is a method of finding a number that is close enough to the correct answer, usually within a specified range.