Square Root of 131

The square root is the inverse of the square of the number. 131 is not a perfect square. The square root of 131 is expressed in both radical and exponential form. In the radical form, it is expressed as √131, whereas (131)^(1/2) in exponential form. √131 ≈ 11.4455, 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 and approximation methods 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. Now let us look at how 131 is broken down into its prime factors:

Step 1: Finding the prime factors of 131 Breaking it down, 131 is a prime number, so its prime factorization is simply 131.

Step 2: Since 131 is not a perfect square, it cannot be grouped into pairs.

Therefore, calculating √131 using prime factorization is not feasible.

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. In the case of 131, we group it as 31 and 1.

Step 2: Now we need to find n whose square is less than or equal to 1. We can use n = 1 because 1×1 = 1. Now the quotient is 1, and after subtracting 1 from 1, the remainder is 0.

Step 3: Bring down 31, which is the new dividend. Add the old divisor with itself, 1 + 1, to get 2, which will be our new divisor.

Step 4: We now have 2n as the new divisor. We need to find the value of n.

Step 5: Find 2n × n ≤ 31. Let n = 1, then 2×1×1 = 2.

Step 6: Subtract 2 from 31; the difference is 29, and the quotient is 11.

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. The new dividend is 2900.

Step 8: Find the new divisor, which is 114 because 229 × 4 = 916.

Step 9: Subtract 916 from 2900 to get the result 1984.

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 until the remainder is zero.

So the square root of √131 is approximately 11.45.

The approximation method is another method for finding 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 131 using the approximation method.

Step 1: Find the closest perfect squares of √131. The smallest perfect square less than 131 is 121, and the largest perfect square greater than 131 is 144. √131 falls somewhere between 11 and 12.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula: (131 - 121) / (144 - 121) = 10 / 23 ≈ 0.435 Add the decimal approximation to the integer part: 11 + 0.435 = 11.435.

So the square root of 131 is approximately 11.44.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, since 4 × 4 = 16.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction (p/q) where p and q are integers and q ≠ 0. For example, √2 is irrational.
   
• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. For example, 131 is a prime number.
   
• Long division method: A method used to find the square root of non-perfect squares by repeated division.
   
• Approximation method: A technique for estimating the square root of a number by finding nearby perfect squares and interpolating between them.