Square Root of 10.3

The square root is the inverse of a square of a number. 10.3 is not a perfect square. The square root of 10.3 is expressed in both radical and exponential form. In radical form, it is expressed as √10.3, whereas (10.3)^(1/2) in exponential form. √10.3 ≈ 3.209, 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:

• Long division method
• Approximation method

The long division method is particularly used for non-perfect square numbers. Let us now learn how to find the square root using the long division method, step by step:

Step 1: Consider the number 10.3 and express it as 10.30 for ease of calculation. Group the digits from right to left in pairs, starting from the decimal point.

Step 2: Find the largest number whose square is less than or equal to 10. The largest square less than 10 is 3 × 3 = 9. Thus, 3 is the first digit of the square root.

Step 3: Subtract 9 from 10 to get 1. Bring down 30 to make it 130.

Step 4: Double the divisor (which is 3), giving 6, and use it to form a new divisor.

Step 5: Find a digit x such that 6x × x is less than or equal to 130. Here, x is 2, because 62 × 2 = 124.

Step 6: Subtract 124 from 130 to get 6. Bring down two zeros to make it 600.

Step 7: Double the current result (32), giving 64, and use it to form a new divisor.

Step 8: Find a digit y such that 64y × y is less than or equal to 600. Here, y is 0, because 640 × 0 = 0.

Step 9: Continue the process until the desired decimal places are obtained.

So the square root of √10.3 ≈ 3.209.

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 10.3 using the approximation method.

Step 1: Identify the closest perfect squares around 10.3. The smallest perfect square less than 10.3 is 9, and the largest perfect square greater than 10.3 is 16. √10.3 falls between √9 = 3 and √16 = 4.

Step 2: Use the formula for approximation: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Step 3: Applying the formula: (10.3 - 9) / (16 - 9) = 1.3 / 7 ≈ 0.186

Step 4: Add this value to the smaller perfect square root: 3 + 0.186 = 3.186

Thus, the approximate square root of 10.3 is 3.209.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: √4 = 2.
   
• Irrational number: An irrational number cannot be written as a simple fraction. Its decimal goes on forever without repeating. Example: √2.
   
• Approximation: An estimation of a value that is nearly but not exactly correct. Approximations are often used for irrational numbers.
   
• Decimal: A fractional number expressed in a system based on powers of ten. Example: 3.14.
   
• Long division method: A step-by-step approach used to find the square root of numbers, especially non-perfect squares, by division.