Square Root of 2.1

The square root is the inverse of the square of the number. 2.1 is not a perfect square. The square root of 2.1 is expressed in both radical and exponential form. In the radical form, it is expressed as √2.1, whereas (2.1)^(1/2) in the exponential form. √2.1 = 1.44914, 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 long-division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. However, since 2.1 is not a perfect square and is not an integer, prime factorization is not applicable here. Therefore, calculating 2.1 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 consider the number 2.1 as 210 (by multiplying by 100 to avoid decimals). We will later adjust for this multiplication.

Step 2: Now we need to find a number whose square is close to 2. Since 1^2 = 1 and 2^2 = 4, we choose 1.

Step 3: Subtract 1 from 2, the remainder is 1.

Step 4: Bring down two zeros to make it 100. The new divisor is double the previous quotient: 2.

Step 5: Find a digit n such that 2n × n ≤ 100. The closest value is n = 4, as 24 × 4 = 96.

Step 6: Subtract 96 from 100, the remainder is 4.

Step 7: Since the dividend is less than the divisor, add a decimal point and bring down 00, making it 400.

Step 8: The new divisor is 28 (from 24 + 4n), and n = 1 works since 281 × 1 = 281.

Step 9: Subtract 281 from 400 to get the remainder 119. Step 10: The quotient is 1.41. Continue to get more decimal places if needed. So, the square root of 2.1 is approximately 1.449.

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. Now let us learn how to find the square root of 2.1 using the approximation method.

Step 1: Now we have to find the closest perfect square of √2.1. The closest perfect squares of 2.1 are 1 (√1 = 1) and 4 (√4 = 2). √2.1 falls between 1 and 2.

Step 2: Now we need to apply the interpolation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (2.1 - 1) / (4 - 1) = 1.1 / 3 = 0.3667 Add this value to the lower square root value: 1 + 0.3667 = 1.3667. Refining further, we find that √2.1 is approximately 1.449.

• Square root: A square root is the inverse of squaring a number. Example: 3^2 = 9, and the inverse is the square root, √9 = 3.
   
• 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.
   
• Decimal number: A number that has a whole part and a fractional part separated by a decimal point, such as 2.1.
   
• Approximation: The process of finding a value that is close enough to the correct answer, typically with the use of rounding or estimation.
   
• Long division method: A technique used for dividing large numbers and for finding square roots of non-perfect squares by dividing the number into smaller, more manageable parts.