Square Root of 110

The square root is the inverse of the square of the number. 110 is not a perfect square. The square root of 110 is expressed in both radical and exponential form.

In the radical form, it is expressed as √110, whereas (110)^(1/2) in the exponential form. √110 ≈ 10.4881, 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, for non-perfect square numbers, 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 110 is broken down into its prime factors.

Step 1: Finding the prime factors of 110 Breaking it down, we get 2 × 5 × 11.

Step 2: Now we found out the prime factors of 110. The second step is to make pairs of those prime factors. Since 110 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 110 using prime factorization is impossible.

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 110, we need to group it as 10 and 1.

Step 2: Now we need to find n whose square is ≤1. We can say n as ‘1’ because 1 × 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 10, 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: Now we get 2n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 10. Let us consider n as 4, now 2 × 4 × 4 = 32, which is more than 10. So, we take n as 3, then 2 × 3 × 3 = 18.

Step 6: Subtract 10 from 18, and the difference is -8, but since we can't have a negative remainder, review steps to ensure the closest n is chosen correctly.

Step 7: Add a decimal point to the quotient and bring down two zeros to the remainder, now making it 1000.

Step 8: The new divisor becomes 26 (2n + n = 23, add another n = 26). Choose n as 3, then 263 × 3 = 789.

Step 9: Subtracting 789 from 1000 gives us 211.

Step 10: Continue this process until you achieve the desired decimal precision.

The square root of √110 is approximately 10.4881.

The approximation method is another method for finding the 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 110 using the approximation method.

Step 1: Now we have to find the closest perfect square to √110. The smallest perfect square less than 110 is 100, and the largest perfect square greater than 110 is 121. √110 falls somewhere between 10 and 11.

Step 2: Use the formula (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square).

Using the formula (110 - 100) ÷ (121 - 100) = 10 ÷ 21 ≈ 0.476. Adding this to the smaller perfect square root, 10 + 0.476 = 10.476.

Thus, the approximate square root of 110 is 10.476.

• 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.

• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is why it is also known as a principal square root.

• Approximation: Approximation is a method used to find an approximate value that is close to the actual number, often used for non-perfect squares.

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