Square Root of 99.9

The square root is the inverse of the square of the number. 99.9 is not a perfect square. The square root of 99.9 is expressed in both radical and exponential form.

In the radical form, it is expressed as √99.9, whereas (99.9)^(1/2) in exponential form. √99.9 ≈ 9.994998749, 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: -

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 99.9 is broken down into its prime factors.

Step 1: Finding the prime factors of 99.9 Breaking it down, we get 2 x 3 x 3 x 5 x 5 x 11.1: 2^1 x 3^2 x 5^2 x 11.1^1

Step 2: Now we found out the prime factors of 99.9. The second step is to make pairs of those prime factors. Since 99.9 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 99.9 using prime factorization is not straightforward.

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 99.9, we need to first consider 99 and 0.9.

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 x 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 90, which is the new dividend (considering 0.9 as 90 for decimal adjustment). Add the old divisor with the same number 9 + 9, we get 18, which will be our new divisor.

Step 4: We now need to find a digit, say ‘m', such that 18m x m is less than or equal to 1890. After calculation, 189 x 9 = 1701.

Step 5: Subtract 1701 from 1890; the difference is 189, and the quotient is 9.9

Step 6: Since the dividend is less than the divisor, we need to add another decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 18900.

Step 7: Continue the process to get more decimal places as needed. Thus, the square root of √99.9 ≈ 9.994998749.

The approximation method is another method for finding the square roots, and 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 99.9 using the approximation method.

Step 1: Now we have to find the closest perfect square of √99.9. The smallest perfect square less than 99.9 is 81, and the largest perfect square greater than 99.9 is 100. √99.9 falls somewhere between 9 and 10.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (99.9 - 81) / (100 - 81) = 18.9 / 19 = 0.994736842.

Adding this decimal to 9 (the integer part), we get 9 + 0.994736842 ≈ 9.994998749, so the square root of 99.9 is approximately 9.995.

• Square root: A square root is the inverse of squaring a number. 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.

• Decimal: If a number has a whole number and a fractional part in a single representation, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.

• Long division method: A method used to find the square root of numbers, especially non-perfect squares, by dividing and averaging progressively.