Square Root of 1082

The square root is the inverse of the square of the number. 1082 is not a perfect square. The square root of 1082 is expressed in both radical and exponential form. In the radical form, it is expressed as √1082, whereas (1082)^(1/2) in the exponential form. √1082 ≈ 32.89377, 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 the long-division method and approximation method 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. Now let us look at how 1082 is broken down into its prime factors:

Step 1: Finding the prime factors of 1082 Breaking it down, we get 2 x 541: 2^1 x 541^1

Step 2: Now we found the prime factors of 1082. The second step is to make pairs of those prime factors. Since 1082 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1082 using prime factorization is not feasible for finding a square root.

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 1082, we need to group it as 82 and 10.

Step 2: Now we need to find n whose square is ≤ 10. We can say n as '3' because 3 x 3 = 9, which is lesser than 10. Now the quotient is 3, and after subtracting 9 from 10, the remainder is 1.

Step 3: Now let us bring down 82, which is the new dividend. Add the old divisor with the same number 3 + 3, we get 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 6n x n ≤ 182. Let us consider n as 2; now 62 x 2 = 124.

Step 6: Subtract 124 from 182; the difference is 58, and the quotient is 32.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeros to the dividend. Now the new dividend is 5800.

Step 8: Now we need to find the new divisor, which is 659, because 659 x 9 = 5931.

Step 9: Subtracting 5931 from 5800, we get the result -131.

Step 10: Now the quotient is 32.89.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.

So the square root of √1082 ≈ 32.89.

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 1082 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √1082. The smallest perfect square less than 1082 is 1024, and the largest perfect square greater than 1082 is 1089. √1082 falls somewhere between 32 and 33.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Going by the formula (1082 - 1024) ÷ (1089 - 1024) = 58 ÷ 65 ≈ 0.892. Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 32 + 0.892 = 32.892.

So the square root of 1082 is approximately 32.892.

• Square root: A square root is the inverse of squaring a number. For example, 4² = 16, and the inverse of the square is the square root, that is, √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, 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, it is usually the positive square root that is used in practical applications. This is known as the principal square root.
   
• Factors: Factors of a number are the numbers that divide it exactly without leaving a remainder. For example, factors of 1082 are 1, 2, 541, and 1082.
   
• Long division method: A method used for finding the square root of non-perfect squares by dividing the number into parts and calculating step by step.