Square Root of 108

The square root is the inverse of the square of the number. 108 is not a perfect square. The square root of 108 is expressed in both radical and exponential form.

In the radical form, it is expressed as √108, whereas (108)^(1/2) in the exponential form. √108 = 10.3923, 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 108 is broken down into its prime factors.

Step 1: Finding the prime factors of 108 Breaking it down, we get 2 x 2 x 3 x 3 x 3: 2² x 3³

Step 2: Now we have found the prime factors of 108. The second step is to make pairs of those prime factors. Since 108 is not a perfect square, therefore the digits of the number can’t be grouped in pairs completely.

Therefore, calculating √108 using prime factorization involves taking the square root of the pairs and leaving one factor unpaired.

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 108, we need to group it as 08 and 1.

Step 2: Now we need to find n whose square is 1. We can say n as ‘1’ because 1 x 1 is lesser than or equal to 1. Now the quotient is 1, and after subtracting 1-1, the remainder is 0.

Step 3: Now let us bring down 08, which is the new dividend. Add the old divisor with the same number, 1 + 1, we get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 08. Let us consider n as 4, now 2 x 4 x 4 = 64.

Step 6: Subtract 08 from 64; the difference is 44, and the quotient is 10.4.

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 4400.

Step 8: Now we need to find the new divisor, which is 9, because 209 x 9 = 1881.

Step 9: Subtracting 1881 from 4400, we get the result 2519.

Step 10: Now the quotient is 10.39.

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

So the square root of √108 is 10.39.

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 108 using the approximation method.

Step 1: Now we have to find the closest perfect square of √108. The smallest perfect square less than 108 is 100 and the largest perfect square greater than 108 is 121. √108 falls somewhere between 10 and 11.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square) Going by the formula (108 - 100) ÷ (121 - 100) = 0.38

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 10 + 0.38 = 10.38

so the square root of 108 is approximately 10.38.

• Square root: A square root is the inverse of a square. For example: 4² = 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 the principal square root.

• Prime factorization: Breaking down a number into its prime factors. For instance, the prime factorization of 108 is 2 x 2 x 3 x 3 x 3.

• 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.