Square Root of 1252

The square root is the inverse of squaring a number. 1252 is not a perfect square. The square root of 1252 is expressed in both radical and exponential form. In radical form, it is expressed as √1252, whereas in exponential form it is expressed as (1252)^(1/2). √1252 ≈ 35.37983, 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, for non-perfect square numbers, the prime factorization method is not suitable, and methods like long division and approximation are used. Let's explore these methods:

• Prime factorization method
• Long division method
• Approximation method

The prime factorization of a number is the product of its prime factors. Now, let's look at how 1252 is broken down into its prime factors.

Step 1: Finding the prime factors of 1252 Breaking it down, we get 2 x 2 x 313: 2^2 x 313^1

Step 2: We have found the prime factors of 1252. Since 1252 is not a perfect square, the prime factors cannot be grouped into pairs.

Therefore, calculating √1252 using prime factorization alone is not possible.

The long division method is particularly useful for non-perfect square numbers. In this method, we find the closest perfect square number to the given number. Let's learn how to find the square root using the long division method, step by step.

Step 1: To begin, group the numbers from right to left. For 1252, group it as 52 and 12.

Step 2: Find n whose square is less than or equal to 12. Here, n is 3 because 3 x 3 = 9, which is less than 12. The quotient is 3, and the remainder is 12 - 9 = 3.

Step 3: Bring down 52, making the new dividend 352. Add the old divisor (3) to itself to form the new divisor, 6.

Step 4: Find n such that 6n x n is less than or equal to 352. Let n = 5. Then, 65 x 5 = 325.

Step 5: Subtract 325 from 352, giving a remainder of 27. The quotient becomes 35.

Step 6: Since the dividend is less than the divisor, add a decimal point and bring down 00, making the new dividend 2700.

Step 7: Find n such that 700n x n is less than or equal to 2700. Let n = 3. Then, 703 x 3 = 2109.

Step 8: Subtract 2109 from 2700, giving a remainder of 591.

Step 9: Continue this process until you achieve the desired precision.

The square root of √1252 is approximately 35.38.

The approximation method is another way to find square roots quickly. Let's find the square root of 1252 using this method.

Step 1: Identify the closest perfect squares around 1252. The smallest perfect square less than 1252 is 1225, and the largest is 1296. Thus, √1252 falls between 35 and 36.

Step 2: Use interpolation to approximate: (1252 - 1225) / (1296 - 1225) = 27 / 71 ≈ 0.38. Adding this decimal to the lower integer gives 35 + 0.38 = 35.38, so the square root of 1252 is approximately 35.38.

• Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the inverse is the square root, √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction. It has a non-repeating, non-terminating decimal expansion.
   
• Long division method: A step-by-step approach to finding square roots, especially for non-perfect squares, involving division and approximation.
   
• Approximation method: A method used to find a close estimate of the square root of a number by identifying nearby perfect squares and interpolating between them.
   
• Interpolation: A mathematical method used to estimate values between two known values, often used in approximation methods for finding square roots.