Square Root of 982

The square root is the inverse of the square of the number. 982 is not a perfect square. The square root of 982 is expressed in both radical and exponential form. In radical form, it is expressed as √982, whereas in exponential form as (982)^(1/2). √982 ≈ 31.3209, 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 prime factorization method is not used; instead, 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 982 is broken down into its prime factors.

Step 1: Finding the prime factors of 982. Breaking it down, we get 2 × 491. Since 491 is a prime number, the factorization cannot form pairs.

Therefore, calculating √982 using prime factorization is not feasible for an exact 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 982, we need to group it as 82 and 9.

Step 2: Now we need to find n whose square is ≤ 9. We can say n is ‘3’ because 3 × 3 = 9, which is equal to 9. Now the quotient is 3, and after subtracting 9 - 9, the remainder is 0.

Step 3: Now let us bring down 82, which is the new dividend. Add the old divisor with the same number: 3 + 3 = 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 × n ≤ 82. Let us consider n as 1, now 6 × 1 × 1 = 61.

Step 6: Subtract 82 from 61; the difference is 21, and the quotient is 31.

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 zeroes to the dividend. Now the new dividend is 2100.

Step 8: Now we need to find the new divisor that is 62 because 622 × 2 = 1244.

Step 9: Subtracting 1244 from 2100, we get the result 856.

Step 10: Now the quotient is 31.2.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero. So the square root of √982 is approximately 31.32.

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 982 using the approximation method.

Step 1: Now we have to find the closest perfect square of √982. The smallest perfect square less than 982 is 961, and the largest perfect square greater than 982 is 1024. √982 falls somewhere between 31 and 32.

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: (982 - 961) ÷ (1024 - 961) = 21 ÷ 63 = 0.3333. 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 31 + 0.33 = 31.33,

so the square root of 982 is approximately 31.33.

• Square root: A square root is the inverse of a square. 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, 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, it is always the positive square root that is more commonly used due to its practical applications. This is known as the principal square root.

• Prime factorization: It is the process of expressing a number as the product of its prime factors. For example, the prime factorization of 18 is 2 × 3 × 3.

• Long division method: A step-by-step process of dividing to find more precise roots of non-perfect squares, especially used for calculating square roots.