Square Root of 984

The square root is the inverse operation of squaring a number. Since 984 is not a perfect square, its square root is expressed in both radical and exponential forms. In radical form, it is √984, while in exponential form, it is expressed as (984)^(1/2). The approximate value of √984 is 31.368, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.

The prime factorization method works well for perfect squares. For non-perfect squares like 984, other methods such as the long division method and approximation method are used. Let's explore these methods: -

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

Prime factorization breaks a number down into its prime factors. Let's examine how 984 is broken down:

Step 1: Finding the prime factors of 984 Breaking it down, we get 2 x 2 x 2 x 3 x 41: 2³ x 3 x 41

Step 2: After identifying the prime factors of 984, we attempt to pair them. Since 984 is not a perfect square, the digits cannot be completely paired, making it impossible to calculate using prime factorization alone.

The long division method is particularly useful for non-perfect square numbers. Here’s how to use it for finding the square root of 984:

Step 1: Group the number from right to left. For 984, we group it as 98 and 4.

Step 2: Find n whose square is less than or equal to 98. n=9 because 9^2=81, which is less than 98. The quotient is 9, and the remainder is 17 after subtracting 81 from 98.

Step 3: Bring down the next pair, 4, making the new dividend 174. Add the last divisor (9) to itself, making it 18.

Step 4: Find a digit which when added to 180 and multiplied by the same digit gives a product less than or equal to 1744. That digit is 9, because 189 x 9 = 1701.

Step 5: Subtract 1701 from 1744, resulting in 43.

Step 6: Since the remainder is less than the divisor, add a decimal point and continue the process.

Step 7: Continue these steps until the desired decimal accuracy is reached. The square root of 984 is approximately 31.368.

The approximation method is a straightforward way to find square roots. Here's how to approximate the square root of 984:

Step 1: Identify the closest perfect squares. 961 (31²) and 1024 (32²) are the nearest perfect squares to 984. Thus, √984 is between 31 and 32.

Step 2: Use interpolation to find a more precise value: (984 - 961) / (1024 - 961) = 23 / 63 ≈ 0.365 Add the result to 31, giving 31 + 0.365 = 31.365

Thus, √984 is approximately 31.365.

• - Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 16 is 4, because 4 × 4 = 16. -

• Irrational number: A number that cannot be written as a simple fraction; its decimal form is non-terminating and non-repeating. Example: √2. -

• Prime factorization: The expression of a number as a product of its prime numbers. -

• Perfect square: A number that is the square of an integer. Example: 49, because 7 × 7 = 49. -

• Long division method: A step-by-step process of dividing numbers to find the square root of non-perfect squares.