Square Root of 3.1

The square root is the inverse of the square of the number. 3.1 is not a perfect square. The square root of 3.1 is expressed in both radical and exponential form. In the radical form, it is expressed as √3.1, whereas (3.1)^(1/2) in exponential form. √3.1 ≈ 1.76068, 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:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Since 3.1 is a decimal and not a perfect square, prime factorization is not applicable in this context. Therefore, calculating 3.1 using prime factorization is impractical.

The long division method is particularly used for non-perfect square numbers. Let us now learn how to find the square root using the long division method, step by step.

Step 1: Start by pairing the digits from the decimal point. For 3.1, treat it as 3.10.

Step 2: Find a number whose square is closest to 3. We can say n is '1' because 1 × 1 ≤ 3. Now the quotient is 1, and the remainder is 2.

Step 3: Bring down the next pair, making it 210. Double the quotient (1), giving us 2, and consider it as the new divisor prefix.

Step 4: Find the largest digit n such that 2n × n ≤ 210. Here, n is '7' because 27 × 7 = 189.

Step 5: Subtract 189 from 210, leaving a remainder of 21.

Step 6: Bring down another pair of zeroes, making it 2100.

Step 7: Double the current quotient (17) to get 34 as the new divisor prefix.

Step 8: Find n such that 34n × n ≤ 2100. Here, n is '6' because 346 × 6 = 2076.

Step 9: Subtract 2076 from 2100, leaving a remainder of 24.

Step 10: The quotient now reads 1.76, which is the approximate square root of 3.1.

The approximation method is another way to find the square roots, providing an easy approach to estimate the square root of a given number. Now let us learn how to find the square root of 3.1 using the approximation method.

Step 1: Identify the closest perfect squares to 3.1. The smallest perfect square is 1, and the largest is 4. √3.1 falls between 1 and 2.

Step 2: Apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (3.1 - 1) / (4 - 1) = 0.7 Add this decimal to the lower perfect square root, which is 1. 1 + 0.7 = 1.7

Step 3: Further refine this to get more precise, using trial and error to find the decimal part, resulting in approximately 1.76068.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, which 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 usually the positive square root that is emphasized due to its uses in the real world. This is also known as the principal square root.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.

• Long division method: This is a method used to find more accurate square roots of non-perfect squares through a systematic process, involving dividing the number into pairs of digits, finding divisors, and iteratively refining the quotient.