Square Root of 997

The square root is the inverse of squaring a number. 997 is not a perfect square. The square root of 997 is expressed in both radical and exponential forms. In radical form, it is expressed as √997, whereas (997)^(1/2) in exponential form. √997 ≈ 31.575, 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 used. Let us now learn the following methods:

1. Prime factorization method
2. Long division method
3. Approximation method

The prime factorization of a number is the product of its prime factors. Now, let us look at how 997 is broken down into its prime factors.

Step 1: Finding the prime factors of 997

997 is a prime number itself, and thus cannot be broken down further. As it is not a perfect square, calculating the square root using prime factorization is not feasible.

The long division method is particularly used 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, group the numbers from right to left. For 997, it can be grouped as 9 and 97.

Step 2: Find n whose square is ≤ 9. We can take n as 3 because 3 × 3 = 9. Subtracting gives a remainder of 0, and the quotient is 3.

Step 3: Bring down 97, making the new dividend 97.

Step 4: Add the old divisor with the same number: 3 + 3 = 6, which will be our new divisor.

Step 5: The new divisor will be 6n. Find n such that 6n × n ≤ 97. Choose n as 1, since 61 × 1 = 61.

Step 6: Subtract 61 from 97 to get 36, and the quotient is 31.

Step 7: Since the dividend is less than the divisor, add a decimal point and bring down two zeros to make the new dividend 3600.

Step 8: The new divisor is 631, as 631 × 5 = 3155.

Step 9: Subtracting 3155 from 3600 gives the result 445.

Step 10: The quotient is 31.5.

Step 11: Continue these steps until reaching the desired decimal places. So, the square root of √997 is approximately 31.575.

The approximation method is another way to find 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 997 using the approximation method.

Step 1: Identify the perfect squares closest to √997. The perfect square smaller than 997 is 961, and the perfect square larger than 997 is 1024. √997 falls between 31 and 32.

Step 2: Use the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (997 - 961) / (1024 - 961) = 36 / 63 ≈ 0.57.

Adding this to the lower bound of the square root: 31 + 0.57 = 31.57. Therefore, the square root of 997 is approximately 31.57.

• Square Root: A square root is the inverse of squaring a number. Example: 4² = 16, and the inverse of squaring is the square root, √16 = 4.

• Irrational Number: An irrational number is one that cannot be expressed in the form p/q, where q is not equal to zero and p and q are integers.

• Prime Number: A prime number is a number greater than 1 with no divisors other than 1 and itself. For example, 2, 3, 5, 7, 11, etc.

• Long Division Method: A procedure used to find the square root of non-perfect squares by dividing the number into pairs and solving iteratively.

• Decimal: If a number has both a whole number and a fractional part, it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.