Square Root of 2.2

The square root is the inverse of the square of the number. 2.2 is not a perfect square. The square root of 2.2 is expressed in both radical and exponential form. In the radical form, it is expressed as √2.2, whereas (2.2)¹/² in the exponential form. √2.2 ≈ 1.48324, 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 applicable for non-perfect square numbers where long-division method and approximation method are used. Let us now 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: To begin with, consider the number 2.2 and write it as 2.2000 for easier calculation.

Step 2: Find a number whose square is less than or equal to 2. The number is 1 because 1² = 1, which is less than 2. The quotient is 1, and after subtracting 1 from 2, the remainder is 1.

Step 3: Bring down 20, making the new dividend 120. Double the divisor 1, giving 2, which will be part of the new divisor.

Step 4: Find a digit n such that 2n × n ≤ 120. The suitable n is 4, because 24 × 4 = 96.

Step 5: Subtract 96 from 120, the difference is 24. Bring down the next pair of zeros, making it 2400.

Step 6: Double the current quotient 14, giving 28. Find a digit n such that 28n × n ≤ 2400. The suitable n is 8, because 288 × 8 = 2304.

Step 7: Subtract 2304 from 2400, the remainder is 96.

Step 8: Continue this process until the desired decimal places are achieved.

Therefore, √2.2 ≈ 1.483.

The approximation method is another way to find 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 2.2 using the approximation method.

Step 1: Identify the perfect squares closest to 2.2. The closest smaller perfect square is 1 (1²), and the closest larger perfect square is 4 (2²). √2.2 lies between 1 and 2.

Step 2: Apply linear interpolation: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (2.2 - 1) / (4 - 1) = 0.4.

Step 3: Add this value to the lower bound of the interval: 1 + 0.4 = 1.4. This is a rough approximation.

Step 4: Refine this approximation using further iterations or more sophisticated methods for better accuracy, leading to √2.2 ≈ 1.483.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: The square root of 4.84 is 2.2.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where p and q are integers and q is not equal to zero.
   
• Principal square root: The principal square root is the non-negative square root of a number. It is typically the one used in calculations.
   
• Interpolation: A method of estimating values between two known values using a linear or non-linear approach.
   
• Decimal: A decimal is a number that includes a fractional part, separated from the integer part by a decimal point. For example, 2.2 and 1.48324 are decimals.