Square Root of 10.8

The square root is the inverse of the square of a number. 10.8 is not a perfect square. The square root of 10.8 is expressed in both radical and exponential forms. In the radical form, it is expressed as √10.8, whereas (10.8)^(1/2) is the exponential form. √10.8 ≈ 3.2863, 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, for non-perfect square numbers, 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 prime factorization method involves expressing a number as a product of prime factors. However, since 10.8 is not a perfect square and is a decimal, prime factorization is not suitable. We will use other methods like long division and approximation for non-integers.

The long division method is particularly used for non-perfect square numbers. Below are the steps to find the square root of 10.8 using the long division method:

Step 1: Pair the digits from right to left. For 10.8, treat it as 1080 for pairing by adding a zero.

Step 2: Find the largest number whose square is less than or equal to the first pair (10). The largest number is 3 since 3 x 3 = 9.

Step 3: Subtract 9 from 10, giving a remainder of 1, and bring down the next pair, making it 18.

Step 4: Double the quotient (3), which is 6, and use it as the next divisor's first digit.

Step 5: Determine a digit (n) such that 6n x n is less than or equal to 180. Here n is 2 because 62 x 2 = 124.

Step 6: Subtract 124 from 180 to get the remainder 56.

Step 7: Add a decimal point to the quotient and bring down two zeros, making the dividend 5600.

Step 8: Double the current quotient part (32), resulting in 64, and find a digit n such that 64n x n ≤ 5600.

Step 9: The process continues similarly to obtain two decimal places for the square root value.

Thus, the square root of 10.8 is approximately 3.2863.

The approximation method is another way to find the square roots. It's a simple method to estimate the square root of a given number. Here's how to find the square root of 10.8 using the approximation method:

Step 1: Identify the closest perfect squares around 10.8. The smallest perfect square is 9, and the largest is 16. Therefore, √10.8 is between 3 and 4.

Step 2: Use linear interpolation to approximate: (10.8 - 9) / (16 - 9) = (x - 3) / (4 - 3) Solving gives x ≈ 3.2863.

Thus, the approximate value of √10.8 is 3.2863.

• Square root: A square root is the inverse of a square. 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 as a fraction of two integers.
   
• Decimal: A decimal consists of a whole number and a fraction represented together, such as 3.2863.
   
• Long division method: A systematic approach to finding a square root by dividing the number into pairs of digits.
   
• Approximation method: Estimating the square root of a number by comparing it to nearby perfect squares and using interpolation.