Square Root of 131.75

The square root is the inverse of the square of a number. 131.75 is not a perfect square. The square root of 131.75 is expressed in both radical and exponential form. In radical form, it is expressed as √131.75, whereas (131.75)^(1/2) in exponential form. √131.75 ≈ 11.478, 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, the 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, we need to group the numbers from right to left. For 131.75, we consider it as 131 and 75.

Step 2: Now we need to find n whose square is less than or equal to 1. We can say n is ‘1’ because 1 x 1 is less than or equal to 1. Now the quotient is 1, and after subtracting 1-1, the remainder is 0.

Step 3: Now let us bring down 31, which is the new dividend. Add the old divisor with the same number: 1 + 1, we get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 31. Let us consider n as 5; now 2 x 5 x 5 = 25.

Step 6: Subtract 31 from 25; the difference is 6, and the quotient is 15.

Step 7: 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. Now the new dividend is 6750.

Step 8: Now we need to find the new divisor that is 109 because 109 x 6 = 654.

Step 9: Subtracting 654 from 675, we get the result 21.

Step 10: Now the quotient is 11.4.

Step 11: Continue doing these steps until we get two numbers after the decimal point. If there are no decimal values, continue till the remainder is zero.

So the square root of √131.75 is approximately 11.48.

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Let us learn how to find the square root of 131.75 using the approximation method.

Step 1: We have to find the closest perfect squares of √131.75. The smallest perfect square less than 131.75 is 121, and the largest perfect square greater than 131.75 is 144. √131.75 falls somewhere between 11 and 12.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula, (131.75 - 121) / (144 - 121) = 0.478. Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 11 + 0.478 = 11.478.

So the square root of 131.75 is approximately 11.478.

• 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 one that cannot be expressed as a simple fraction; its decimal form is non-repeating and non-terminating.
   
• Decimal: A decimal is a number that includes a whole number and a fraction represented with a dot. Examples include 7.86, 8.65, and 9.42.
   
• Approximation: A method used to estimate the value of a number, especially useful for non-perfect squares.
   
• Long division method: A step-by-step process to find the square root of a number by performing division repeatedly.