Square Root of 1005

The square root is the inverse of the square of the number. 1005 is not a perfect square. The square root of 1005 is expressed in both radical and exponential form. In the radical form, it is expressed as √1005, whereas (1005)^(1/2) in the exponential form. √1005 ≈ 31.701, 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, where long-division and approximation methods are used. Let us now learn the following methods: -

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

The product of prime factors is the prime factorization of a number. Now let us look at how 1005 is broken down into its prime factors.

Step 1: Finding the prime factors of 1005 Breaking it down, we get 3 x 5 x 67: (3¹x5¹ x 67¹).

Step 2: Now we have found the prime factors of 1005. The second step is to make pairs of those prime factors. Since 1005 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1005 using prime factorization is not straightforward.

The long division method is particularly used for non-perfect square numbers. This method involves finding 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 1005, we need to group it as 05 and 10.

Step 2: Now we need to find n whose square is less than or equal to 10. We can say n is ‘3’ because \(3 \times 3 = 9\), which is less than 10. Now the quotient is 3, and after subtracting 9 from 10, the remainder is 1.

Step 3: Bring down 05, making the new dividend 105. Double the quotient for the new divisor: 3 + 3 = 6.

Step 4: Find a digit d such that \(6d \times d\) is less than or equal to 105. Trying d = 1, we have \(61 \times 1 = 61\), which is less than 105.

Step 5: Subtract 61 from 105, giving a remainder of 44. Bring down 00, making the new dividend 4400.

Step 6: Double the quotient 31 for the new divisor: 31 + 31 = 62. Find a digit d such that \(62d \times d\) is less than or equal to 4400. Trying d = 7, we have \(627 \times 7 = 4389\).

Step 7: Subtract 4389 from 4400, resulting in a remainder of 11. The quotient so far is 31.7.

Step 8: Continue this process to obtain more decimal places if needed. The square root of 1005 is approximately 31.701.

The approximation method is another method for finding square roots, and it is an easy way to estimate the square root of a given number. Let us learn how to find the square root of 1005 using the approximation method.

Step 1: Find the closest perfect squares to 1005. The smallest perfect square less than 1005 is 961, and the nearest perfect square greater than 1005 is 1024. √1005 falls somewhere between 31 and 32.

Step 2: Apply the formula: (Given number - smaller perfect square) ÷ (larger perfect square - smaller perfect square) (1005 - 961) ÷ (1024 - 961) = 44 ÷ 63 ≈ 0.698

Adding this to the smaller integer root: 31 + 0.698 = 31.698 So the approximate square root of 1005 is 31.698.

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

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

• Principal square root: A number has both positive and negative square roots; however, the positive square root is more prominent due to its applications in the real world. This is known as the principal square root.

• Perfect square: A perfect square is an integer that is the square of another integer. For example: 16 is a perfect square because 4 × 4 = 16.

• Long division method: A mathematical procedure used to find the square root of a non-perfect square by dividing the number into pairs and calculating step by step.