Square Root of 9900

The square root is the inverse of the square of the number. 9900 is not a perfect square. The square root of 9900 is expressed in both radical and exponential form. In the radical form, it is expressed as √9900, whereas (9900)^(1/2) in the exponential form. √9900 ≈ 99.4987, 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 9900 is broken down into its prime factors.

Step 1: Finding the prime factors of 9900

Breaking it down, we get 2 × 2 × 3 × 3 × 5 × 5 × 11: 2^2 × 3^2 × 5^2 × 11

Step 2: Now we found out the prime factors of 9900. The second step is to make pairs of those prime factors. The square root is found by taking one number from each pair of the same number, which yields: 2 × 3 × 5 × √11 = 30√11

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 9900, we need to group it as 99 and 00.

Step 2: Now we need to find n whose square is less than or equal to 99. We can say n is '9' because 9 × 9 = 81, which is less than 99. Now the quotient is 9, after subtracting 81 from 99, the remainder is 18.

Step 3: Now let us bring down 00, 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 the sum of the dividend and quotient. Now we get 18n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 18n × n ≤ 1800. Let us consider n as 9, now 18 × 9 = 162.

Step 6: Subtract 1620 from 1800; the difference is 180, and the quotient is 99.

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 18000.

Step 8: Now we need to find the new divisor; let's consider 189 because 1890 × 9 = 17010.

Step 9: Subtracting 17010 from 18000, we get the result 990.

Step 10: Now the quotient is 99.4.

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

So the square root of √9900 is approximately 99.50.

The approximation method is another method for finding the 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 9900 using the approximation method.

Step 1: Now we have to find the closest perfect square of √9900. The smallest perfect square less than 9900 is 9801, and the largest perfect square more than 9900 is 10000. √9900 falls somewhere between 99 and 100.

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 (9900 - 9801) / (10000 - 9801) = 99/199 = 0.4975

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 99 + 0.4975 = 99.4975, so the square root of 9900 is approximately 99.50.

• 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.
   
• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 9900 is 2 × 2 × 3 × 3 × 5 × 5 × 11.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 100 is a perfect square because it is 10^2.
   
• Exponent: A mathematical notation indicating the number of times a number is multiplied by itself. For example, in 5^3, the exponent is 3, indicating 5 is multiplied by itself three times.