Square Root of 46.1

The square root is the inverse of the square of the number. 46.1 is not a perfect square. The square root of 46.1 is expressed in both radical and exponential form.

In the radical form, it is expressed as √46.1, whereas (46.1)^(1/2) in the exponential form. √46.1 ≈ 6.791, 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. Since 46.1 is not a perfect square, using prime factorization directly is not feasible. Therefore, calculating the square root of 46.1 using prime factorization is not possible.

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 pair up the digits from right to left. In the case of 46.1, we consider 46 and 10.

Step 2: Now we need to find n whose square is closest to 46. We can say n is 6 because 6 × 6 = 36, which is less than 46. Now the quotient is 6, and after subtracting 36 from 46, the remainder is 10.

Step 3: Now let us bring down 10, making the new dividend 1010. Add the old divisor with the same number: 6 + 6 = 12, making it our new divisor.

Step 4: The new divisor, 12n, is adjusted to find n such that 12n × n ≤ 1010. Let us consider n as 8, as 128 × 8 = 1024, which exceeds 1010. So, n should be 7, making 127 × 7 = 889.

Step 5: Subtract 889 from 1010; the difference is 121.

Step 6: Since the remainder exists, we need to add a decimal point and continue the process by bringing down pairs of zeros.

Step 7: Continue this process until you have two decimal places of accuracy.

The square root of 46.1 is approximately 6.791.

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 46.1 using the approximation method.

Step 1: Now we have to find the closest perfect square of √46.1. The smallest perfect square below 46.1 is 36 (6^2) and the largest perfect square above 46.1 is 49 (7^2). Therefore, √46.1 is between 6 and 7.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula: (46.1 - 36) / (49 - 36) = 10.1 / 13 = 0.7769 Adding this to the lower bound, we get 6 + 0.7769 ≈ 6.777.

Therefore, the square root of 46.1 is approximately 6.791 after further refinement.

• 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 has more prominence due to its uses in the real world. This 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.
   
• Long division method: A method used to find the square roots of non-perfect square numbers through a series of divisions and approximations.