Square Root of 9700

The square root is the inverse of the square of the number. 9700 is not a perfect square. The square root of 9700 is expressed in both radical and exponential form. In the radical form, it is expressed as √9700, whereas (9700)^(1/2) in the exponential form. √9700 ≈ 98.489, 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 9700 is broken down into its prime factors:

Step 1: Finding the prime factors of 9700

Breaking it down, we get 2 x 2 x 5 x 5 x 97: 2^2 x 5^2 x 97

Step 2: Now we found out the prime factors of 9700. The second step is to make pairs of those prime factors. Since 9700 is not a perfect square, therefore the digits of the number can’t be grouped in complete pairs. Therefore, calculating 9700 using prime factorization can only provide an estimate.

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 9700, we need to group it as '00' and '97'.

Step 2: Now we need to find n whose square is less than or equal to 97. We can say n as ‘9’ because 9 x 9 = 81 is less than 97. Now the quotient is 9, and after subtracting 81 from 97, the remainder is 16.

Step 3: Now let us bring down '00', making it 1600, which is the new dividend. Add the old divisor with the same number 9 + 9, we get 18, which will be our new divisor.

Step 4: The new divisor will be 18n, and we now need to find a digit n such that 18n x n is less than or equal to 1600.

Step 5: The next step is finding 18n x n ≤ 1600. Let us consider n as 8. So, 188 x 8 = 1504.

Step 6: Subtract 1504 from 1600; the difference is 96, and the quotient becomes 98.

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 zeros to the dividend. Now the new dividend is 9600.

Step 8: Now we need to find the new divisor. The number 985 fits since 985 x 5 = 4925.

Step 9: Subtracting 4925 from 9600, we get the result 4675.

Step 10: Now the quotient is 98.4.

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 √9700 ≈ 98.49.

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 9700 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √9700. The smallest perfect square less than 9700 is 9604 (98^2) and the largest perfect square greater than 9700 is 9801 (99^2). √9700 falls somewhere between 98 and 99.

Step 2: Now we need to apply the formula that is: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula: (9700 - 9604) / (9801 - 9604) = 96 / 197 ≈ 0.487

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 98 + 0.487 = 98.487, so the square root of 9700 is approximately 98.487.

• 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, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.
   
• Prime factorization: It involves breaking down a number into its basic prime factors. Example: 9700 = 2^2 x 5^2 x 97.


• 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, and 9.42 are decimals.