Square Root of 40.8

The square root is the inverse of the square of the number. 40.8 is not a perfect square. The square root of 40.8 is expressed in both radical and exponential form.

In radical form, it is expressed as √40.8, whereas (40.8)^(1/2) in exponential form. √40.8 ≈ 6.385, 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, for non-perfect square numbers, the long-division method and approximation method are often used. Let us 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 40.8 is broken down into its prime factors. However, since 40.8 is not a whole number, we convert it to a fraction 408/10 and factor each part.

Step 1: Finding the prime factors of 408

Breaking it down, we get 2 x 2 x 2 x 3 x 17: 2^3 x 3^1 x 17^1

Step 2: Now we found out the prime factors of 408. The second step is to make pairs of those prime factors. Since 40.8 is not a perfect square, the digits of the number can’t be grouped in pairs evenly. Therefore, calculating 40.8 using prime factorization is more complex and not straightforward.

The long division method is particularly useful for non-perfect square numbers. 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 40.8, we consider it as 40.80.

Step 2: Now we need to find n whose square is less than or equal to 40. The closest perfect square is 36, so n is 6. The quotient is 6, and the remainder is 4.8 after subtracting 36 from 40.8.

Step 3: Bring down 00 to get 480. Add the old divisor with the same number 6 + 6 to get 12, which will be our new divisor prefix.

Step 4: The new divisor will be 12n. We need to find n such that 12n × n ≤ 480. Let us consider n as 3, now 123 × 3 = 369.

Step 5: Subtract 369 from 480 to get the remainder 111.

Step 6: Since the dividend is less than the divisor, add a decimal point. Bringing down two zeroes gives 11100.

Step 7: The new divisor is 126. We find n as 8 because 1268 × 8 = 10144.

Step 8: Subtracting 10144 from 11100 gives a remainder of 956.

Step 9: The quotient is approximately 6.38. Step 10: Continue these steps until the desired decimal precision is achieved.

Therefore, the square root of √40.8 ≈ 6.385.

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Let us learn how to find the square root of 40.8 using the approximation method.

Step 1: We find the closest perfect squares around √40.8.

The smallest perfect square below 40.8 is 36, and the largest above is 49. Thus, √40.8 falls between 6 and 7.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) (40.8 - 36) / (49 - 36) = 4.8 / 13 ≈ 0.369 Adding this to the lower perfect square root: 6 + 0.369 = 6.369, so the square root of 40.8 is approximately 6.369.

• 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, the positive square root, known as the principal square root, is often used due to its prominence in real-world applications.
   
• Decimal: A decimal is a number that includes a whole number and a fractional part, represented by digits after a decimal point. For example, 7.86 and 8.65 are decimals.
   
• Long Division Method: A method used to find the square root of non-perfect squares by dividing the number into groups and performing division iteratively to approximate the root.