Square Root of 10.6

The square root is the inverse of the square of the number. 10.6 is not a perfect square. The square root of 10.6 is expressed in both radical and exponential form. In the radical form, it is expressed as √10.6, whereas (10.6)^(1/2) in the exponential form. √10.6 ≈ 3.255, 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 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. However, 10.6 is a decimal and not suitable for prime factorization directly in the traditional sense. Therefore, calculating the square root of 10.6 using prime factorization is not applicable.

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 consider the number 10.6.

Step 2: Find the largest integer whose square is less than or equal to 10.6. In this case, 3^2 = 9 is close to 10.6.

Step 3: Subtract 9 from 10.6 to get 1.6. Bring down pairs of zeros to make it 160.

Step 4: Double the current quotient (3) to get 6, which will be the starting point of the new divisor.

Step 5: Determine a digit X such that 6X * X is less than or equal to 160. The digit is 2 because 62 * 2 = 124.

Step 6: Subtract 124 from 160 to get 36.

Step 7: Bring down another pair of zeros to get 3600.

Step 8: Continue this process to achieve the desired precision for the square root.

The square root of 10.6 is approximately 3.255.

The approximation method is another method for finding the square roots; it is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 10.6 using the approximation method.

Step 1: Find the closest perfect square numbers around 10.6. Here, 9 and 16 are the closest perfect squares. So, √10.6 falls between 3 and 4.

Step 2: Use linear approximation to find that (10.6 - 9)/(16 - 9) = 0.229.

Step 3: Add this decimal to the lower bound of our range (3). So, 3 + 0.229 = 3.229, a rough estimation of the square root.

Step 4: Refine the answer using more decimal places if necessary.

The square root of 10.6 is approximately 3.255.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 × 4 = 16.
   
• Irrational number: An irrational number cannot be written in the form of p/q, where q ≠ 0 and p and q are integers.
   
• Decimal number: A decimal number includes a whole number and a fraction represented with a decimal point, for example, 10.6.
   
• Long division method: A method used to find more precise values of square roots for non-perfect squares.
   
• Approximation: Estimating a number's value through various methods to get a close estimation, often used for square roots.