Square Root of 1.1

The square root is the inverse of squaring a number. 1.1 is not a perfect square. The square root of 1.1 is expressed in both radical and exponential form. In the radical form, it is expressed as √1.1, whereas (1.1)^(1/2) in the exponential form. √1.1 ≈ 1.0488, 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 like 1.1, methods such as the long division method and approximation method are used. Let us learn the following methods:

• Long division method
   
• Approximation method

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: Group the numbers from right to left. For 1.1, consider it as 11 (ignoring the decimal for initial steps).

Step 2: Find n whose square is less than or equal to 1. We can say n is '1' because 1 × 1 ≤ 1. Now the quotient is 1, and after subtracting 1 - 1, the remainder is 0.

Step 3: Bring down the next digit, which is 1, to make the new dividend 10.

Step 4: The new divisor is twice the quotient from step 2, which is 2.

Step 5: Determine n such that 2n × n ≤ 10. n is 4, as 2 × 4 × 4 = 8.

Step 6: Subtract 8 from 10 to get a remainder of 2.

Step 7: Add a decimal point to the quotient and bring down a pair of zeroes to make the new dividend 200.

Step 8: The new divisor is 28 (from 24 + 4).

Step 9: Determine n such that 28n × n ≤ 200. n is 7, as 28 × 7 = 196.

Step 10: Subtract 196 from 200 to get a remainder of 4.

Step 11: Continue this process to obtain more decimal places if necessary.

So the square root of √1.1 ≈ 1.0488.

The approximation method is another method for finding square roots; it is an easy method to estimate the square root of a given number. Let us learn how to find the square root of 1.1 using the approximation method.

Step 1: Identify two perfect squares between which 1.1 falls. The closest perfect squares are 1 (1^2) and 1.21 (1.1^2).

Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). Using the formula: (1.1 - 1) / (1.21 - 1) ≈ 0.476.

Step 3: The approximate square root is 1 + 0.0488 = 1.0488. So the square root of 1.1 is approximately 1.0488.

• Square root: A square root is the inverse operation of squaring a number. For example, 2² = 4, and the square root of 4 is √4 = 2.

• Irrational number: An irrational number is a number that cannot be precisely expressed as a fraction p/q, where p and q are integers and q ≠ 0.

• Decimal: A decimal is a number expressed in the base-10 numeral system, which includes a whole number and fractional part separated by a decimal point, such as 1.0488.

• Long division method: A procedure used to divide two numbers to obtain a quotient and remainder. It's also used to find square roots of non-perfect squares.

• Approximation method: A technique used to find an approximate value of irrational numbers by identifying nearby perfect squares and estimating based on them.