Square Root of 371

The square root is the inverse operation of squaring a number. 371 is not a perfect square. The square root of 371 can be expressed in both radical and exponential forms. In radical form, it is expressed as √371, whereas in exponential form it is written as 371^(1/2). The square root of 371 is approximately 19.26136, which is an irrational number because it cannot be expressed as a fraction of two integers.

For non-perfect square numbers like 371, methods such as the long division method and approximation method are employed, instead of the prime factorization method used for perfect squares. Let's explore these methods:

• Long division method 
• Approximation method

The long division method is commonly used for finding square roots of non-perfect squares. Here's how it's done step by step for 371:

Step 1: Group the numbers from right to left. For 371, consider it as 371.

Step 2: Find the largest number whose square is less than or equal to the group. Here, 5^2 = 25, and 6^2 = 36. Use 6 as the divisor because 36 is the closest perfect square less than 37. The quotient is 6, and the remainder is 1 (37 - 36 = 1).

Step 3: Bring down the next pair of digits, making the new dividend 171.

Step 4: Double the quotient (6), resulting in 12. Use 12 as the beginning of the new divisor, and find a digit n such that 12n * n is less than or equal to 171.

Step 5: The closest value for n is 4, since 124 * 4 = 496. Subtract 496 from 1710 to get the remainder 54.

Step 6: Add a decimal point to the quotient and pair two zeros to the remainder, making it 5400.

Step 7: Repeat the process with new dividends and divisors until the desired decimal precision is achieved.

The square root of 371 is approximately 19.26136.

The approximation method provides a quick way to find the square root by identifying the closest perfect squares surrounding the number.

Step 1: Identify the perfect squares closest to 371.

The perfect squares around 371 are 361 (19^2) and 400 (20^2).

So, √371 is between 19 and 20.

Step 2: Use interpolation to approximate the decimal value.

The formula is:

(Given number - smaller perfect square) / (larger perfect square - smaller perfect square).

For 371, it is: (371 - 361) / (400 - 361) = 10 / 39 ≈ 0.256

Step 3: Add this fractional part to the lower square root: 19 + 0.256 ≈ 19.26

Thus, the square root of 371 is approximately 19.26.

• Square root: A square root is the inverse operation of squaring a number. For example, 5^2 = 25, and the square root of 25 is √25 = 5.
   
• Irrational number: An irrational number is a number that cannot be written as a simple fraction, with non-repeating and non-terminating decimal expansion.
   
• Principal square root: The principal square root is the non-negative square root of a number. For example, the principal square root of 25 is 5, not -5.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because it is 6^2.
   
• Long division method: A method used to find the square root of non-perfect squares by performing a series of divisions and approximations.