Square Root of 401

The square root is the inverse of the square of a number. 401 is not a perfect square. The square root of 401 is expressed in both radical and exponential form. In radical form, it is expressed as √401, whereas in exponential form, it is (401)^(1/2). √401 ≈ 20.02498, which is an irrational number because it cannot be expressed in the form p/q, where p and q are integers and q ≠ 0.

The prime factorization method can be used for perfect square numbers. However, for non-perfect square numbers, the long division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

Prime factorization involves expressing a number as the product of its prime factors. Since 401 is a prime number, it cannot be broken down into other prime factors. Therefore, calculating the square root of 401 using prime factorization is not possible.

The long division method is particularly useful for non-perfect square numbers. Here's how to find the square root of 401 using the long division method, step by step:

Step 1: Group the digits of 401 from right to left. In the case of 401, we have one group of three digits: 401.

Step 2: Find the largest number whose square is less than or equal to 401. The closest perfect square is 400, and its square root is 20.

Step 3: Subtract the square of 20 from 401, which gives a remainder of 1. The quotient is 20.

Step 4: Bring down a pair of zeroes to make the dividend 100.

Step 5: Double the quotient (20) which gives us a new divisor of 40.

Step 6: Find a digit n such that 40n × n is less than or equal to 100. The suitable n is 2, as 402 × 2 = 80.

Step 7: Subtract 80 from 100, leaving a remainder of 20.

Step 8: Continue the process by bringing down more pairs of zeroes and repeating the steps above to get more decimal places in the quotient.

The square root of 401 is approximately 20.02498.

The approximation method is another way to find square roots. It is an easy method for estimating the square root of a number. Here's how to approximate the square root of 401:

Step 1: Identify the closest perfect squares around 401.

The closest are 400 (20²) and 441 (21²).

√401 falls between 20 and 21.

Step 2: Use the formula:

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

For 401, this is (401 - 400) / (441 - 400) = 1 / 41 ≈ 0.02439.

Step 3: Add this result to the smaller root: 20 + 0.02439 ≈ 20.02439.

Therefore, the square root of 401 is approximately 20.02439.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, 4² = 16, so √16 = 4.
   
• Irrational number: An irrational number cannot be written as a simple fraction, as it has a non-repeating and non-terminating decimal expansion.
   
• Prime number: A prime number has only two factors—1 and itself.
   
• Approximation: This refers to finding a value that is close to but not exactly equal to the actual value.
   
• Long division method: A technique used to divide large numbers and can be adapted to find square roots of non-perfect squares.