Square Root of 19.2

The square root is the inverse of the square of the number. 19.2 is not a perfect square. The square root of 19.2 is expressed in both radical and exponential form. In the radical form, it is expressed as √19.2, whereas (19.2)^(1/2) in the exponential form. √19.2 ≈ 4.38178, 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. Now let us look at how 19.2 is broken down into its prime factors.

Step 1: Finding the prime factors of 19.2

Since 19.2 is not a whole number, we first multiply it by 10 to get 192.

The prime factorization of 192 is 2 x 2 x 2 x 2 x 2 x 3.

Step 2: Now we found out the prime factors of 192. The second step is to make pairs of those prime factors. Since 19.2 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Thus, calculating 19.2 using prime factorization is complex without further simplification.

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 19.2, we need to group it as 19 and 20 (19.2 is treated as 1920 for convenience).

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

Step 3: Bring down the next group, which is 20, to make it 320. Add the old divisor with the same number: 4 + 4 = 8, which will be our new divisor.

Step 4: The new divisor is 8n. Find n such that 8n × n ≤ 320. Let us consider n as 3, now 83 × 3 = 249.

Step 5: Subtract 320 from 249, the difference is 71, and the quotient is 4.3.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeros to the remainder. Now the new dividend is 7100.

Step 7: Find the new divisor by considering 86 as the new divisor (83 + 3) and find n that satisfies 86n × n ≤ 7100. Suppose n is 8, then 868 × 8 = 6944.

Step 8: Subtract 6944 from 7100, we get the result 156.

Step 9: Continue doing these steps until we reach an appropriate level of precision.

So the square root of √19.2 is approximately 4.38178.

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 19.2 using the approximation method.

Step 1: Now we have to find the closest perfect square of √19.2. The smallest perfect square less than 19.2 is 16, and the largest perfect square more than 19.2 is 25. √19.2 falls somewhere between 4 and 5.

Step 2: Now we need to apply the formula that is

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

Going by the formula (19.2 - 16) ÷ (25 - 16) = 0.3556.

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 + 0.38178 = 4.38178, so the square root of 19.2 is approximately 4.38178.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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 is more prominent due to its uses in the real world, which is why 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.
   
• Prime factorization: The process of determining which prime numbers multiply together to form a given number. For example, the prime factorization of 192 is 2 × 2 × 2 × 2 × 2 × 3.