Square Root of 14.4

The square root is the inverse of the square of a number. 14.4 is not a perfect square. The square root of 14.4 is expressed in both radical and exponential form. In the radical form, it is expressed as √14.4, whereas (14.4)^(1/2) is the exponential form. √14.4 ≈ 3.79473, 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 typically used for non-perfect square numbers, where methods such as 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 14.4 can be expressed in terms of its prime factors.

Step 1: Converting 14.4 into a fraction: 14.4 = 144/10 = (12 x 12)/(2 x 5).

Step 2: The prime factorization of 144 is 2 x 2 x 2 x 2 x 3 x 3. The prime factorization of 10 is 2 x 5.

Step 3: Now, we pair the prime factors of the numerator. Since 14.4 is not a perfect square, calculating √14.4 using prime factorization directly involves approximations.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number to 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 digits in pairs from right to left. In the case of 14.4, consider it as 1440 after multiplying by 100 for ease.

Step 2: Find an integer whose square is less than or equal to 14. Consider n = 3 because 3 x 3 = 9, which is less than 14.

Step 3: Subtract 9 from 14, giving a remainder of 5. Bring down the next pair, making it 540.

Step 4: Double the current quotient (3) to get 6, and use it as the base for the next divisor.

Step 5: Find a digit x such that 6x x x is less than or equal to 540. Using trial, x = 8 works because 68 x 8 = 544.

Step 6: Subtract 544 from 540, resulting in -4, so adjust x to 7, where 67 x 7 = 469.

Step 7: Continue the process to find the quotient to the desired decimal places.

The approximate square root is 3.79473.

The approximation method is another method for finding square roots; it is an easy method to approximate the square root of a given number. Let us learn how to find the square root of 14.4 using the approximation method.

Step 1: Identify the closest perfect squares around 14.4.

The closest smaller perfect square is 9 (√9 = 3), and the closest larger perfect square is 16 (√16 = 4).

Step 2: Using interpolation, approximate between these values: (14.4 - 9) / (16 - 9) = (14.4 - 9) / 7 = 0.7714

Step 3: Approximate the square root: 3 + 0.7714 = 3.7714

Thus, √14.4 ≈ 3.79473 using a more precise calculation.

• Square root: A square root is the inverse of squaring a number. For example, 4² = 16, and the inverse, the square root, is √16 = 4
   
• Irrational number: An irrational number cannot be expressed as a fraction p/q, where q is not zero and p and q are integers.
   
• Approximation: The process of finding a value close to the true value, often used when the exact value is difficult to find.
   
• Long division method: A technique used to find the square root of a non-perfect square number through a series of division steps.
   
• Decimal: A numerical representation that includes a whole number and a fractional part, separated by a decimal point, such as 7.86.