Square Root of 931

The square root is the inverse of the square of the number. 931 is not a perfect square. The square root of 931 is expressed in both radical and exponential form. In the radical form, it is expressed as √931, whereas 931^(1/2) in the exponential form. √931 ≈ 30.52868, 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. Now let us look at how 931 is broken down into its prime factors:

Step 1: Finding the prime factors of 931 Since 931 is not easily factorable into simple primes and is also not a perfect square, the prime factorization is not straightforward.

Therefore, calculating 931 using prime factorization is not practical for finding the square root.

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 931, we need to group it as 31 and 9.

Step 2: Now we need to find n whose square is 9. We can say n as ‘3’ because 3 × 3 is lesser than or equal to 9. Now the quotient is 3, and after subtracting 9-9 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 3 + 3 we get 6 which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 31. Let us consider n as 5, now 65 × 5 = 325.

Step 6: Subtract 31 from 25, the difference is 6, and the quotient is 30.

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 600.

Step 8: Now we need to find the new divisor that is 7 because 607 × 7 = 4249.

Step 9: Subtracting 4249 from 6000 we get the result 1751.

Step 10: Now the quotient is 30.52.

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

So the square root of √931 is approximately 30.53.

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 931 using the approximation method.

Step 1: Now we have to find the closest perfect square of √931.

The smallest perfect square less than 931 is 900 and the largest perfect square greater than 931 is 961.

√931 falls somewhere between 30 and 31.

Step 2: Now we need to apply the formula that is

(Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square).

Going by the formula (931 - 900) ÷ (961-900) = 31/61 ≈ 0.508.

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 30 + 0.508 ≈ 30.508, so the square root of 931 is approximately 30.53.

• 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.
   
• Long division method: A method used to find the square root of non-perfect square numbers by systematically dividing and estimating.
   
• Approximation method: A method used to estimate the square root of a number by finding nearby perfect squares and interpolating between them.
   
• Factors: Numbers that can be multiplied together to produce another number. For example, the factors of 931 include 1, 7, 133, and 931.