Square Root of 397

The square root is the inverse of the square of a number. 397 is not a perfect square. The square root of 397 is expressed in both radical and exponential form. In the radical form, it is expressed as √397, whereas in exponential form, it is (397)^(1/2). √397 ≈ 19.933, 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; instead, long-division and approximation methods 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. Let's see how 397 is broken down into its prime factors.

Step 1: Finding the prime factors of 397 397 is a prime number, so it cannot be factored into smaller prime numbers.

Therefore, calculating 397 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 group the numbers from right to left. In the case of 397, we can use 3 | 97.

Step 2: Now, we need to find a number whose square is less than or equal to 3. The number is 1 because 1 × 1 = 1. Now the quotient is 1, and after subtracting 1 from 3, the remainder is 2.

Step 3: Bring down 97, making the new dividend 297. Double the current quotient and add a digit to form a new divisor. The current quotient is 1, so 2 × 1 = 2.

Step 4: Now find a digit, n, such that 2n × n is less than or equal to 297. Let's consider n as 9, so 29 × 9 = 261.

Step 5: Subtract 261 from 297, which leaves 36. The quotient is now 19.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to continue with the process.

Step 7: Add two zeroes to the dividend, making it 3600.

Step 8: Now find a new divisor. 2 × 19 = 38, and find n such that 38n × n ≤ 3600. Let's consider n as 9, so 389 × 9 = 3501.

Step 9: Subtract 3501 from 3600, which leaves 99.

Step 10: Now the quotient is 19.9.

Step 11: Continue doing these steps until we reach the desired decimal precision.

So the square root of √397 is approximately 19.933.

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's learn how to find the square root of 397 using the approximation method.

Step 1: We have to find the closest perfect squares around √39

The smallest perfect square less than 397 is 361, and the largest perfect square greater than 397 is 400.

√397 falls somewhere between 19 and 20.

Step 2: Now we need to apply the formula:

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

Using the formula (397 - 361) / (400 - 361) = 36 / 39 ≈ 0.923.

Adding the integer part gives us 19 + 0.923 = 19.923, so the square root of 397 is approximately 19.923.

• Square root: A square root is the inverse operation of squaring a number. For example, 5^2 = 25, and the inverse is the square root, √25 = 5.
   
• Irrational number: An irrational number cannot be written as a simple fraction, where the denominator is not zero and both numerator and denominator are integers.
   
• Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. It has exactly two distinct positive divisors: 1 and itself.
   
• Decimal: A decimal number consists of a whole number part and a fractional part separated by a decimal point. For example, 3.14, 7.86, and 19.933 are decimals.
   
• Long division method: A method for dividing two numbers and obtaining a quotient. It's often used for finding square roots of non-perfect squares.