Square Root of 2.4

The square root is the inverse of the square of the number. 2.4 is not a perfect square. The square root of 2.4 is expressed in both radical and exponential form. In the radical form, it is expressed as √2.4, whereas (2.4)^(1/2) in the exponential form. √2.4 ≈ 1.54919, 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 2.4 is broken down into its prime factors.

Step 1: Multiply 2.4 by 10 to get a whole number, resulting in 24.

Step 2: Finding the prime factors of 24. Breaking it down, we get 2 x 2 x 2 x 3: 2^3 x 3^1

Step 3: Now, take the square root of 24 and divide it by the square root of 10 to get the square root of 2.4.

Step 4: Since 24 is not a perfect square, the digits cannot be grouped in pairs.

Therefore, calculating 2.4 using prime factorization directly is not straightforward.

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 multiply 2.4 by 100 to work with whole numbers, resulting in 240.

Step 2: Group the numbers from right to left. In the case of 240, we need to group it as 40 and 2.

Step 3: Find n whose square is less than or equal to 2. We can say n is '1' because 1 x 1 is less than or equal to 2. The quotient is 1, and the remainder is 1.

Step 4: Bring down the next group which is 40, making the new dividend 140. Double the previous quotient (1), which gives us 2 as the new divisor.

Step 5: Find a digit, say 'm', such that 2m x m is less than or equal to 140. Here, m is 5 because 25 x 5 = 125.

Step 6: Subtract 125 from 140; the difference is 15. Bring down two zeros to get 1500.

Step 7: The new divisor is 30 (2 * 5) plus the next digit from the quotient. Continue these steps until you achieve the desired precision.

Thus, the square root of 2.4 is approximately 1.549.

The approximation method is another method for finding the 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 2.4 using the approximation method.

Step 1: Find the closest perfect squares around 2.4. The smallest perfect square is 1 (√1 = 1) and the largest perfect square is 4 (√4 = 2). Thus, √2.4 falls between 1 and 2.

Step 2: Now, apply the formula: (Given number - smallest perfect square) / (greater perfect square - smallest perfect square) (2.4 - 1) / (4 - 1) = 1.4 / 3 ≈ 0.467

Step 3: Add this value to the square root of the smaller perfect square: 1 + 0.467 = 1.467 Thus, the square root of 2.4 is approximately 1.549 when calculated more precisely.

• 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.
   
• 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 mathematical method used to find the square root of non-perfect squares through a series of steps involving division and averaging.
   
• Approximation method: A technique used to find an approximate value of a square root by identifying nearby perfect squares and interpolating between them.