Square Root of 820

The square root is the inverse of the square of the number. 820 is not a perfect square. The square root of 820 is expressed in both radical and exponential form.

In the radical form, it is expressed as √820, whereas (820)^(1/2) in the exponential form. √820 ≈ 28.63564, 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 820 is broken down into its prime factors.

Step 1: Finding the prime factors of 820

Breaking it down, we get 2 x 2 x 5 x 41: 2² x 5¹ x 41¹

Step 2: Now we found out the prime factors of 820. The second step is to make pairs of those prime factors. Since 820 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating √820 using prime factorization is impossible.

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 820, we need to group it as 20 and 8.

Step 2: Now we need to find n whose square is 8. We can say n as ‘2’ because 2 x 2 is less than or equal to 8. Now the quotient is 2, and after subtracting 4 from 8, the remainder is 4.

Step 3: Now let us bring down 20, which is the new dividend. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

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

Step 5: The next step is finding 4n x n ≤ 420. Let us consider n as 6, now 46 x 6 = 276.

Step 6: Subtract 276 from 420, the difference is 144, and the quotient is 26.

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 14400.

Step 8: Now we need to find the new divisor that is 576 because 572 x 5 = 2885.

Step 9: Subtracting 2885 from 14400, we get the result 11515.

Step 10: Now the quotient is 28.635.

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 √820 is approximately 28.63564.

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 820 using the approximation method.

Step 1: Now we have to find the closest perfect square of √820. The smallest perfect square less than 820 is 784, and the largest perfect square less than 820 is 841. √820 falls somewhere between 28 and 29.

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, (820 - 784) ÷ (841-784) = 0.6315789.

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 28 + 0.635 = 28.635, so the square root of 820 is approximately 28.635.

• 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, q is not equal to zero, and p and q are integers.
   
• Long division method: This is a method used to find the square root of non-perfect squares by dividing the number into groups of two digits from right to left and performing division iteratively.
   
• Approximation method: A method used to estimate the square root by identifying nearby perfect squares and calculating the difference.
   
• Prime factorization: The process of expressing a number as the product of its prime factors.