Square Root of 19.6

The square root is the inverse of the square of the number. 19.6 is not a perfect square. The square root of 19.6 is expressed in both radical and exponential form. In the radical form, it is expressed as √19.6, whereas (19.6)^(1/2) in the exponential form. √19.6 ≈ 4.427, 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:

• 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 right to left. In the case of 19.6, we can consider it as 196/10 to simplify.

Step 2: Now we need to find n whose square is less than or equal to 19. We can say n is 4 because 4 × 4 = 16, which is less than 19. Now the quotient is 4, and after subtracting 16 from 19, the remainder is 3.

Step 3: Now, let us bring down 60, making it 360. Add the old divisor with the same number: 4 + 4 = 8, which will be our new divisor.

Step 4: The new divisor will be 8n. We need to find the value of n where 8n × n ≤ 360. Let us consider n as 4, now 84 × 4 = 336.

Step 5: Subtract 336 from 360, and the difference is 24. The quotient becomes 4.4.

Step 6: Since the dividend is less than the divisor, we add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 2400.

Step 7: Now we need to find the new divisor, which is 88 because 884 × 4 = 3536 is more than 2400, so the previous divisor works.

Step 8: Subtract 1760 from 2400 to get the remainder 640. Continue this process until you achieve the desired precision.

The square root of 19.6 is approximately 4.427.

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 19.6 using the approximation method.

Step 1: Now we have to find the closest perfect square of √19.6.

The smallest perfect square less than 19.6 is 16, and the largest perfect square greater than 19.6 is 25.

√19.6 falls somewhere between 4 and 5.

Step 2: Now we need to apply the formula:

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

Using the formula: (19.6 - 16)/(25 - 16) = 0.4

Using the formula, we identified the decimal point of our square root.

The next step is adding the value we got initially to the decimal number which is 4 + 0.4 = 4.4, so the square root of 19.6 is approximately 4.427 when refined further.

• Square root: A square root is the inverse of a square. For 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, q is not equal to zero, and p and q are integers.
   
• Approximation: Approximation is the process of finding a value that is close enough to the correct value, often used for irrational numbers.
   
• 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.
   
• Long division method: A method used to find the square root of non-perfect squares through a step-by-step division process.