Square Root of 251

The square root is the inverse of the square of the number. 251 is not a perfect square. The square root of 251 is expressed in both radical and exponential form. In the radical form, it is expressed as √251, whereas \(251^{1/2}\) in the exponential form. √251 ≈ 15.84298, 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 251 is broken down into its prime factors.

Step 1: Finding the prime factors of 251 251 is a prime number, so it cannot be broken down further.

Step 2: Since 251 is a prime number and not a perfect square, calculating its square root using prime factorization is not applicable.

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 251, we need to group it as 51 and 2.

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

Step 3: Now let us bring down 51, which is the new dividend. Add the old divisor with the same number (\(1 + 1 = 2\)), 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, we need to find the value of n.

Step 5: The next step is finding \(2n \times n ≤ 151\). Let us consider n as 7, now \(27 \times 7 = 189\).

Step 6: Subtract 189 from 151, the difference is -38, and the quotient is 17.

Step 7: Since the remainder is negative, we need to adjust the divisor to 26 and redo the calculation, giving us a new remainder of 32.

Step 8: Adding the decimal point allows us to add two zeros to the dividend. Now the new dividend is 3200.

Step 9: Now we need to find the new divisor, which is 158 because (158 times 15 = 2370).

Step 10: Subtracting 2370 from 3200, we get the result 830.

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

So the square root of √251 is approximately 15.84.

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 251 using the approximation method.

Step 1: Now we have to find the closest perfect square of √251. The smallest perfect square less than 251 is 225, and the largest perfect square greater than 251 is 256. √251 falls somewhere between 15 and 16.

Step 2: Now we need to apply the formula: ((text{Given number} - text{smallest perfect square}) / (text{Greater perfect square} - text{smallest perfect square})). Going by the formula ((251 - 225) ÷ (256 - 225) = 0.84). 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 (15 + 0.84 = 15.84).

So the square root of 251 is approximately 15.84.

• 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, that is √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, q is not equal to zero, and p and q are integers.
   
• Prime number: A prime number is a number greater than 1 that has no divisors other than 1 and itself. Example: 251 is a prime number.
   
• Perfect square: A perfect square is a number that is the square of an integer. Example: 16 is a perfect square because it is (4^2).
   
• Long division method: A method used to find the square root of non-perfect squares by performing division operations repeatedly.