Square Root of 396.25

The square root is the inverse of the square of the number. 396.25 is not a perfect square. The square root of 396.25 is expressed in both radical and exponential form. In the radical form, it is expressed as √396.25, whereas (396.25)^(1/2) in the exponential form. √396.25 ≈ 19.902, 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 396.25 is broken down into its prime factors.

Step 1: Finding the prime factors of 396.25 Breaking down 396.25 is complex due to the decimal, so direct prime factorization is not feasible.

Step 2: Since 396.25 is not a perfect square, we can't group its digits into pairs for prime factorization.

Therefore, calculating √396.25 using prime factorization is not appropriate.

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, considering 39625 (ignoring the decimal for now).

Step 2: Find n whose square is close to 39. Here, n = 6 because 6 x 6 = 36, which is less than 39. The quotient is 6, subtract 36 from 39, and the remainder is 3.

Step 3: Bring down 62 (next pair of digits); the new dividend is 362. Add the old divisor (6) with the same number to get 12, which is the new partial divisor.

Step 4: Find n for which 12n x n ≤ 362. Here, n = 2, since 122 x 2 = 244.

Step 5: Subtract 244 from 362, the difference is 118, and the quotient is 62.

Step 6: Bring down 5 (considering the decimal), making the new dividend 1185.

Step 7: The new divisor becomes 124 (122 plus 2), find n for which 124n x n ≤ 1185. Here, n = 9, since 1249 x 9 = 11241.

Step 8: Subtract 11241 from 11850, resulting in 609.

Step 9: Since you're finding a decimal, continue the process to obtain more decimal places.

The quotient is approximately 19.902.

The approximation method is another method for finding square roots and is an easy method to find the square root of a given number. Now let us learn how to find the square root of 396.25 using the approximation method.

Step 1: Now we have to find the perfect squares closest to √396.25.

The smallest perfect square less than 396.25 is 361, and the largest perfect square is 400.

√396.25 falls between 19 and 20.

Step 2: Apply the formula

(Given number - smaller perfect square) / (larger perfect square - smaller perfect square).

(396.25 - 361) / (400 - 361) ≈ 0.902

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 19 + 0.902 = 19.902.

• 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, 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, the positive square root is more commonly used due to its applications in the real world.
   
• Rational number: A rational number can be expressed as the quotient or fraction of two integers, where the denominator is not zero.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, 9.42 are decimals.