Square Root of 9.8

The square root is the inverse operation of squaring a number. 9.8 is not a perfect square. The square root of 9.8 is expressed in both radical and exponential form. In the radical form, it is expressed as √9.8, whereas in the exponential form it is (9.8)^(1/2). √9.8 ≈ 3.1305, 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 squares like 9.8, approximation methods such as the long-division 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 learn how to find the square root using the long division method, step by step:

Step 1: Start by pairing the digits starting from the decimal point. For 9.8, treat it as 98 under the long division method.

Step 2: Find a number whose square is less than or equal to 9. The closest is 3, since 3 x 3 = 9.

Step 3: Subtract 9 from 9 to get 0, and bring down 80.

Step 4: Double the quotient and bring it down as 6_, looking for a digit that fits.

Step 5: Find a digit 'n' such that (60+n) x n ≤ 80. The suitable n is 1, as 61 x 1 = 61.

Step 6: Subtract 61 from 80 to get 19, then bring down two zeros to get 1900.

Step 7: Repeat the process, finding the next digit and quotient to approximate the square root further.

Approximation method is another way to find square roots and is useful for quick estimates. Let's find the square root of 9.8 using this method:

Step 1: Identify two perfect squares between which 9.8 falls. It lies between 9 (3^2) and 16 (4^2).

Step 2: Use linear approximation to find the decimal: (9.8 - 9) / (16 - 9) = 0.8 / 7 ≈ 0.1143

Step 3: Add this to the lower bound square root: 3 + 0.1143 ≈ 3.1143.

Thus, the square root of 9.8 ≈ 3.1143.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: 3^2=9, so √9=3.
   
• Irrational number: A number that cannot be expressed as a fraction of two integers. For example, √9.8 is irrational.
   
• Approximation: The process of finding a value that is close enough to the right answer, usually within a specified range.
   
• Perfect square: A number that is the square of an integer. For example, 9 is a perfect square since it is 3^2.
   
• Long division method: A technique used to find the square root of numbers, especially non-perfect squares, using a step-by-step division process.