Square Root of 111

The square root is the inverse of the square of the number. 111 is not a perfect square. The square root of 111 is expressed in both radical and exponential form.

In the radical form, it is expressed as √111, whereas in the exponential form it is expressed as (111)^(1/2). √111 ≈ 10.53565, 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. Now let us look at how 111 is broken down into its prime factors.

Step 1: Finding the prime factors of 111 Breaking it down, we get 3 x 37.

Step 2: Now that we have found the prime factors of 111, the next step is to make pairs of those prime factors. Since 111 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 111 using prime factorization is not straightforward.

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, we need to group the numbers from right to left. In the case of 111, we need to group it as 11 and 1.

Step 2: Now we need to find n whose square is 1. We can say n is ‘1’ because 1 x 1 is less than or equal to 1. Now the quotient is 1, after subtracting 1-1, the remainder is 0.

Step 3: Now let us bring down 11, which is the new dividend. Add the old divisor with the same number 1 + 1, we get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 2n x n ≤ 11. Let us consider n as 4; now 2 x 4 x 4 = 8.

Step 6: Subtract 11 from 8; the difference is 3, and the quotient is 10.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 300.

Step 8: Now we need to find the new divisor that is 105 because 205 x 5 = 1025.

Step 9: Subtracting 1025 from 3000, we get the result 1975.

Step 10: Now the quotient is 10.5.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √111 is approximately 10.53.

The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 111 using the approximation method.

Step 1: Now we have to find the closest perfect square of √111. The smallest perfect square less than 111 is 100, and the largest perfect square greater than 111 is 121. √111 falls somewhere between 10 and 11.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (111 - 100) ÷ (121 - 100) = 0.524.

Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 10 + 0.524 = 10.524,

so the square root of 111 is approximately 10.53.

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

• Irrational number: An irrational number is a number that cannot 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: 36 is a perfect square because it is 6².

• Prime factorization: The process of expressing a number as the product of its prime factors.

• Decimal approximation: The representation of a number that is not exact but close enough to the value for practical purposes.