Square Root of 981

The square root is the inverse operation of squaring a number. 981 is not a perfect square. The square root of 981 can be expressed in both radical and exponential form. In radical form, it is expressed as √981, whereas in exponential form as (981)^(1/2). √981 ≈ 31.32092, which is an irrational number because it cannot be expressed as a fraction of two integers where the denominator is not zero.

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, we use methods like the long division method and the approximation method. Let us explore these methods:

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

Prime factorization involves expressing a number as the product of its prime factors. Let us break down 981 into its prime factors:

Step 1: Finding the prime factors of 981 Breaking it down, we get 3 x 3 x 109: 3^2 x 109^1

Step 2: Now that we have found the prime factors of 981, we attempt to form pairs of prime factors. Since 981 is not a perfect square, the digits of the number cannot be grouped into pairs, making it impossible to calculate √981 using prime factorization directly.

The long division method is particularly useful for non-perfect square numbers. This method involves finding a series of approximations to reach the square root. Here is how you can find the square root using the long division method, step by step:

Step 1: Group the digits of the number in pairs from right to left. For 981, we group it as 9 and 81.

Step 2: Find n such that ⁿ² is less than or equal to 9. Here, n is 3 because 3 x 3 = 9. Subtract 9 from 9, and the remainder is 0.

Step 3: Bring down the next pair of digits, 81, making the new dividend 81. Double the quotient obtained in the previous step (3), giving a new divisor of 6.

Step 4: Find a digit x such that 6x multiplied by x is less than or equal to 81. Here x is 1 because 61 x 1 = 61. Subtract 61 from 81 to get a remainder of 20.

Step 5: Bring down two zeros to make the dividend 2000. Double the quotient (31), making the new divisor 62.

Step 6: Find a digit y such that 62y x y is less than or equal to 2000. Here y is 3 because 623 x 3 = 1869. Subtract 1869 from 2000 to get a remainder of 131.

Step 7: Repeat the process to obtain more decimal places.

The quotient so far is 31.32. So the square root of √981 is approximately 31.32.

The approximation method is an easy way to estimate the square roots of numbers. Here's how to approximate the square root of 981:

Step 1: Find the closest perfect squares around 981. The closest perfect squares are 961 (31²) and 1024 (32²). √981 falls between 31 and 32.

Step 2: Apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). (981 - 961) / (1024 - 961) = 20 / 63 ≈ 0.317

Using the formula, the approximate decimal part is 0.317. Adding this to the integer part gives us 31 + 0.317 ≈ 31.317.

Thus, the square root of 981 is approximately 31.317.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: 4² = 16, so the square root of 16 is √16 = 4.

• Irrational number: A number that cannot be expressed as a simple fraction, with a non-repeating, non-terminating decimal expansion.

• Long division method: A step-by-step process used to find the square root of a non-perfect square by dividing and averaging.

• Approximation: An approach to estimating a number or value, often used when an exact calculation is not possible or necessary.

• Perfect square: A number that is the square of an integer. For example, 16 is a perfect square because it is 4^2.