Square Root of 253

The square root is the inverse of the square of the number. 253 is not a perfect square. The square root of 253 is expressed in both radical and exponential form. In the radical form, it is expressed as √253, whereas (253)^(1/2) in the exponential form. √253 ≈ 15.90597, 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 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 253 is broken down into its prime factors:

Step 1: Finding the prime factors of 253 Breaking it down, we get 11 x 23.

Step 2: Since 253 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 253 using prime factorization is not straightforward.

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 253, we need to group it as 53 and 2.

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

Step 3: Now let us bring down 53, which is the new dividend. Add the old divisor with the same number 1 + 1 to 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 ≤ 153. Let us consider n as 7, now 27 x 7 = 189, which is too large, so n is 5.

Step 6: Subtract 125 (25 x 5) from 153; the difference is 28, 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 2800.

Step 8: Now we need to find the new divisor that is 310 because 310 x 9 = 2790. Step 9: Subtracting 2790 from 2800, we get the result 10.

Step 10: Now the quotient is 15.9

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 √253 is approximately 15.90.

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 253 using the approximation method.

Step 1: Now we have to find the closest perfect square of √253. The smallest perfect square less than 253 is 225, and the largest perfect square greater than 253 is 256. √253 falls somewhere between 15 and 16.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (253 - 225) ÷ (256 - 225) = 28 ÷ 31 ≈ 0.903. 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 + 0.903 ≈ 15.903.

So, the square root of 253 is approximately 15.90.

• 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.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example, 16 is a perfect square because it is 4².
   
• Approximation: It refers to finding a value that is close enough to the right answer, usually within a certain tolerance. In the context of square roots, it means finding a decimal value close to the actual square root.
   
• Long division method: A technique used to find the square root of non-perfect squares by dividing the number into pairs of digits, starting from the decimal point, and using a step-by-step division process.