Square Root of 20.2

The square root is the inverse of the square of the number. 20.2 is not a perfect square. The square root of 20.2 is expressed in both radical and exponential form. In the radical form, it is expressed as √20.2, whereas (20.2)^(1/2) in the exponential form. √20.2 ≈ 4.49444, 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 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 group the numbers from the decimal point. In the case of 20.2, consider it as 20.20 and group it as 20 and 20.

Step 2: Now we need to find n whose square is less than or equal to 20. We can say n is ‘4’ because 4 x 4 = 16, which is less than or equal to 20. Now the quotient is 4; after subtracting 16 from 20, the remainder is 4.

Step 3: Bring down 20, making the new dividend 420. Add the old divisor (4) with itself, giving us 8, which will be our new divisor prefix.

Step 4: Find a digit (x) such that 8x multiplied by x is less than or equal to 420. The number 84 fits because 84 x 4 = 336.

Step 5: Subtract 336 from 420, resulting in a remainder of 84. The quotient so far is 4.4.

Step 6: Add a decimal point and two zeros to the remainder to continue. Now the new dividend is 8400.

Step 7: Repeat the process to find the next digit of the quotient, which will be 9 since 849 x 9 = 7641.

Step 8: Subtract 7641 from 8400, giving a remainder of 759.

Step 9: Continuing this process will yield a more precise square root.

The square root of √20.2 is approximately 4.494.

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 20.2 using the approximation method.

Step 1: Identify the closest perfect squares around 20.2.

The smallest perfect square less than 20.2 is 16, and the largest perfect square more than 20.2 is 25.

Therefore, √20.2 falls between 4 and 5.

Step 2: Apply the linear approximation formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using this, (20.2 - 16) / (25 - 16) = 4.2 / 9 ≈ 0.4667.

Now add this decimal to the lower bound (4): 4 + 0.4667 ≈ 4.47.

Thus, the approximate square root of 20.2 is about 4.47.

• 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 in the form of p/q, where q is not equal to zero, and p and q are integers.
   
• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.
   
• Linear approximation: A method to estimate the value of a function close to a known value, often used to approximate square roots by finding the nearest perfect squares.