Square Root of 19600

The square root is the inverse of the square of the number. 19600 is a perfect square. The square root of 19600 is expressed in both radical and exponential form. In the radical form, it is expressed as √19600, whereas (19600)^(1/2) in the exponential form. √19600 = 140, which is a rational number because it can be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method can be used for perfect square numbers. However, methods like the long division method and approximation are used for non-perfect square numbers. Let us now learn the following methods: -

1. Prime factorization method 
2. Long division method 
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 19600 is broken down into its prime factors.

Step 1: Finding the prime factors of 19600 Breaking it down, we get 2 x 2 x 2 x 2 x 5 x 5 x 7 x 7: 2⁴ x 5² x 7²

Step 2: Now we have found the prime factors of 19600. The second step is to make pairs of those prime factors. Since 19600 is a perfect square, the digits of the number can be grouped in pairs.

Therefore, calculating 19600 using prime factorization gives us √19600 = 2² x 5 x 7 = 140.

The long division method is particularly used for finding square roots. 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 19600, we need to group it as 00 and 196.

Step 2: Now we need to find n whose square is closest to 196. We can say n is '14' because 14 x 14 = 196.

Step 3: The quotient is 14, and since there is no remainder, the square root of 19600 is exactly 140.

The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. However, since 19600 is a perfect square, the exact square root is 140.

Students can make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in methods. Now let us look at a few of those mistakes that students tend to make in detail.

• Square root: A square root is the inverse of a square. Example: 12² = 144, and the inverse of the square is the square root, that is, √144 = 12.

• Rational number: A rational number is a number that can 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. This is why it is also known as the principal square root.

• Perfect square: A perfect square is a number that is the square of an integer. Example: 19600 is a perfect square because it is 140².

• Long division method: A systematic method used to find the square root of a number by dividing it into groups and solving step by step.