Square Root of 111.68

The square root is the inverse of the square of the number. 111.68 is not a perfect square. The square root of 111.68 is expressed in both radical and exponential form.

In the radical form, it is expressed as √111.68, whereas (111.68)^(1/2) in the exponential form. √111.68 ≈ 10.56817, 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:

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

The product of prime factors is the prime factorization of a number. However, since 111.68 is not an integer, we cannot directly use prime factorization. Therefore, calculating 111.68 using prime factorization is inappropriate.

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, group the digits of 111.68 from right to left as 11|16|80.

Step 2: Find the largest number whose square is less than or equal to 11, which is 3 (since 3^2 = 9). Subtract 9 from 11 to get a remainder of 2, and bring down 16 to make it 216.

Step 3: Double the divisor (3), making it 6, and find a digit x such that 6x × x ≤ 216. The suitable x is 3, since 63 × 3 = 189. Subtract 189 from 216 to get 27, and bring down 80 to make it 2780.

Step 4: Double the current divisor (63), making it 126, and find a digit y such that 126y × y ≤ 2780. The suitable y is 2, since 1262 × 2 = 2524. Subtract 2524 from 2780 to get 256.

Step 5: Continue the process to get more decimal places.

The square root of 111.68 is approximately 10.568.

The approximation method is another method for finding square roots, and it is an easy method to find the square root of a given number. Let us learn how to find the square root of 111.68 using the approximation method:

Step 1: Identify the perfect squares nearest to 111.68, which are 100 (10^2) and 121 (11^2). Therefore, √111.68 falls between 10 and 11.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). For 111.68: (111.68 - 100) / (121 - 100) = 11.68 / 21 ≈ 0.556.

Step 3: Add this decimal value to the smaller root: 10 + 0.556 = 10.556.

Therefore, the square root of 111.68 is approximately 10.556.

• Square root: A square root is the inverse operation of squaring a number. For example, if 5² = 25, then the square root of 25 is 5.

• Irrational number: An irrational number is a number that cannot be written as a simple fraction; its decimal representation is non-repeating and non-terminating.

• Radical: A symbol (√) used to denote the square root or nth root of a number.

• Decimal: A number that consists of a whole number part and a fractional part separated by a decimal point, such as 10.568.

• Approximation: The process of finding a value close to the true value, often used when dealing with irrational numbers or complex calculations.