Square Root of 13.1

The square root is the inverse of the square of the number. 13.1 is not a perfect square. The square root of 13.1 is expressed in both radical and exponential form. In the radical form, it is expressed as √13.1, whereas (13.1)^(1/2) in exponential form. √13.1 ≈ 3.619, which is an irrational number because it cannot be expressed as a fraction 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 and approximation methods 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, we group the numbers from right to left. In the case of 13.1, we consider it as 13.10 for easy calculation.

Step 2: Now we need to find n whose square is less than or equal to 13. We can say n is ‘3’ because 3 × 3 = 9, which is less than 13. Now the quotient is 3, after subtracting 13 - 9, the remainder is 4.

Step 3: Bring down 10 to make the new dividend 40. Add the old divisor with the same number (3 + 3 = 6) to get the new divisor.

Step 4: The new divisor is now 6n. We need to find a value of n such that 6n × n ≤ 40. Let us consider n as 6, now 66 × 6 = 396.

Step 5: Subtract 40 from 36, the difference is 4, and the quotient is 3.6.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. The new dividend is 400.

Step 7: Find the new divisor, which is 72 because 72 × 5 = 360.

Step 8: Subtract 360 from 400 to get the result 40.

Step 9: The quotient is 3.61, continue these steps until we get two numbers after the decimal point. If no decimal values exist, continue until the remainder is zero.

So the square root of √13.1 is approximately 3.619.

The approximation method is another method for finding square roots; it is an easy way to find the square root of a given number. Now let us learn how to find the square root of 13.1 using the approximation method.

Step 1: Find the closest perfect squares of √13.1.

The smallest perfect square less than 13.1 is 9, and the largest perfect square greater than 13.1 is 16. √13.1 falls somewhere between 3 and 4.

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

Using the formula (13.1 - 9) ÷ (16 - 9) = 4.1 ÷ 7 ≈ 0.586.

Step 3: Add the initial integer part to the decimal: 3 + 0.586 ≈ 3.586, so the square root of 13.1 is approximately 3.619.

• 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, the positive square root is often used due to its applications in the real world. This is known as the principal square root.
   
• Long division method: A step-by-step process of dividing numbers to find square roots of non-perfect squares.
   
• Approximation method: A technique to estimate the square root of a number by comparing it to nearby perfect squares.