Square Root of 9800

The square root is the inverse of squaring a number. 9800 is not a perfect square. The square root of 9800 is expressed in both radical and exponential form. In the radical form, it is expressed as √9800, whereas in exponential form it is (9800)^(1/2). The square root of 9800 is 98.994949, 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 the 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 9800 is broken down into its prime factors.

Step 1: Finding the prime factors of 9800

Breaking it down, we get 2 x 2 x 2 x 5 x 5 x 7 x 7: 2^3 x 5^2 x 7^2

Step 2: Now that we have found the prime factors of 9800, the next step is to make pairs of those prime factors. Since 9800 is not a perfect square, its digits can’t be grouped in perfect pairs to give an integer square root.

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 9800, we need to group it as 98 and 00.

Step 2: Now we need to find n whose square is less than or equal to 98. We can say n is ‘9’ because 9 x 9 = 81 which is lesser than 98. Now the quotient is 9 after subtracting 98 - 81, the remainder is 17.

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

Step 4: The new divisor will be 18n. We need to find n such that 18n × n ≤ 1700.

Step 5: The next step is finding 18n × n ≤ 1700. Let us consider n as 9, now 189 x 9 = 1701, which is slightly above 1700, so we choose n = 8. Thus, 188 x 8 = 1504.

Step 6: Subtract 1504 from 1700, the difference is 196, and the quotient is 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 zeroes to the dividend. Now the new dividend is 19600.

Step 8: Now we need to find the new divisor that is 198 because 1980 ✖ 9 = 17820.

Step 9: Subtracting 17820 from 19600, we get the result 1780.

Step 10: The quotient is now 98.9.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose there is no decimal value, continue till the remainder is zero.

So the square root of √9800 is approximately 98.99.

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 9800 using the approximation method.

Step 1: Now we have to find the closest perfect square of √9800. The smallest perfect square less than 9800 is 9604 (98^2) and the largest perfect square greater than 9800 is 10000 (100^2). √9800 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). Using this formula: (9800 - 9604) / (10000 - 9604) = 196 / 396 = 0.4949.

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 + 0.4949 = 98.4949, so the square root of 9800 is approximately 98.99.

• Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the inverse operation is the square root, √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, but the positive square root is often used in practical applications and is known as the principal square root.
   
• Prime factorization: A method of expressing a number as the product of its prime factors. For example, the prime factorization of 9800 is 2^3 × 5^2 × 7^2.
   
• Long division method: A method used to find the square roots of non-perfect square numbers through a systematic division process.