Square Root of 10.2

The square root is the inverse of the square of the number. 10.2 is not a perfect square. The square root of 10.2 is expressed in both radical and exponential form. In the radical form, it is expressed as √10.2, whereas (10.2)^(1/2) in the exponential form. √10.2 ≈ 3.194, 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:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the Prime factorization of a number. However, since 10.2 is not a perfect square and not an integer, prime factorization is not applicable for finding its square root. We rely on other methods like long division or approximation for such numbers.

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: To begin with, we need to pair the digits. For 10.2, we consider 10 and 20 (considering the decimal).

Step 2: Find the largest number whose square is less than or equal to 10. This number is 3 (as 3^2 = 9). Subtract 9 from 10 to get a remainder of 1.

Step 3: Bring down 20, making it 120. Double the 3 to get 6, and find a number which when placed next to 6 and multiplied with the same number gives a product less than or equal to 120.

Step 4: The number is 1 because 61 x 1 = 61. Subtract 61 from 120 to get 59.

Step 5: Add a decimal point and bring down two zeros to make it 5900.

Step 6: The next divisor will be 62x, where x is the next digit in the quotient.

Step 7: Continue the process to find more decimal places as needed.

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 10.2 using the approximation method.

Step 1: Identify the nearest perfect squares around 10.2. The closest squares are 9 (3^2) and 16 (4^2).

Step 2: Since 10.2 is closer to 9, we know √10.2 is slightly greater than 3.

Step 3: Use linear approximation between 3 and 4: (10.2 - 9) / (16 - 9) = (10.2 - 9) / 7 ≈ 0.171

Step 4: Add this to 3 to get approximately 3.171, which is close to the actual square root.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. Example: 3.194² ≈ 10.2.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction; its decimal goes on forever without repeating.
   
• Principal square root: The principal square root is the non-negative square root of a number.
   
• Decimal: A decimal is a number that consists of a whole number and a fractional part, separated by a decimal point.
   
• Approximation: Approximating a value involves finding a value that is close to the exact amount, usually due to the impossibility of expressing the exact value.