Square Root of 252

The square root is the inverse of the square of the number. 252 is not a perfect square. The square root of 252 is expressed in both radical and exponential form. In radical form, it is expressed as √252, whereas (252)^(1/2) in exponential form. √252 ≈ 15.8745, 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 252 is broken down into its prime factors.

Step 1: Finding the prime factors of 252 Breaking it down, we get 2 x 2 x 3 x 3 x 7: 2^2 x 3^2 x 7

Step 2: Now we have found out the prime factors of 252. The second step is to make pairs of those prime factors. Since 252 is not a perfect square, the digits of the number can’t be grouped in complete pairs. Therefore, calculating √252 using prime factorization will involve taking square roots of the pairs and multiplying by the remaining factor.

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 252, we need to group it as 52 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 1-1, the remainder is 1.

Step 3: Now let us bring down 52, 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; we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 152; let us consider n as 7, now 27 x 7 = 189

Step 6: Subtract 152 from 189; since it doesn't fit, we adjust n to 6, making 26 x 6 = 156. Subtracting 156 from 152 gives a negative value, so we adjust again to 5, making 25 x 5 = 125 with remainder 27.

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 2700.

Step 8: Now we need to find the new divisor that is 315 because 315 x 8 = 2520.

Step 9: Subtracting 2520 from 2700, we get the result 180.

Step 10: Now the quotient is approximately 15.8.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Continue until the remainder is zero.

So the square root of √252 is approximately 15.874

Approximation method is another method for finding the 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 252 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √252. The smallest perfect square less than 252 is 225 and the largest perfect square greater than 252 is 256. √252 falls somewhere between 15 and 16.

Step 2: Now we need to apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). Using the formula (252 - 225) ÷ (256 - 225) = 27 ÷ 31 ≈ 0.87 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 + 0.87 = 15.87, so the square root of 252 is approximately 15.87.

• 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.
   
• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is why it is also known as the principal square root.
   
• Prime factorization: The process of breaking down a number into its smallest prime factors. For example, the prime factorization of 252 is 2² x 3² x 7.
   
• Long division method: A step-by-step approach to find the square root of a number by dividing it into parts, estimating, and refining the estimate to find an accurate square root.