Square Root of 819

The square root is the inverse of the square of the number. 819 is not a perfect square. The square root of 819 is expressed in both radical and exponential form.

In the radical form, it is expressed as √819, whereas (819)^(1/2) in the exponential form. √819 ≈ 28.6007, 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 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 819 is broken down into its prime factors.

Step 1: Finding the prime factors of 819

Breaking it down, we get 3 x 3 x 7 x 13: 3² x 7 x 13

Step 2: Now we have found the prime factors of 819. The second step is to make pairs of those prime factors. Since 819 is not a perfect square, the digits of the number can’t be grouped in a complete pair. Therefore, calculating √819 using prime factorization directly is not possible.

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 819, we need to group it as 19 and 8.

Step 2: Now we need to find n whose square is ≤ 8. We can say n is ‘2’ because 2 x 2 = 4, which 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 19, making the new dividend 419. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

Step 4: We now have 4n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 4n x n ≤ 419. Let us consider n as 9, now 49 x 9 = 441, which is greater than 419, so we choose n as 8, making 48 x 8 = 384.

Step 6: Subtract 384 from 419, and the difference is 35. The quotient is 28.

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, making it 3500.

Step 8: Now we need to find the new divisor, which is 576, because 576 x 6 = 3456.

Step 9: Subtract 3456 from 3500, and we get the result 44.

Step 10: Now the quotient is 28.6

Step 11: Continue these steps until we get two numbers after the decimal point. If there are no decimal values, continue till the remainder is zero.

So the square root of √819 is approximately 28.60.

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 819 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √819. The smallest perfect square less than 819 is 784, and the largest perfect square greater than 819 is 841. √819 falls somewhere between 28 and 29.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (819 - 784) / (841 - 784) = 35 / 57 ≈ 0.614

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 + 0.614 = 28.614, so the square root of 819 is approximately 28.614.

• 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 usually the positive square root that is emphasized due to its practical applications. That is why it is also known as the principal square root.
   
• Integers: The combination of whole numbers and negative numbers are integers, for example: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, etc., are integers.
   
• Decimal: If a number has a whole number and a fraction represented in a single number, then it is called a decimal, for example: 7.86, 8.65, and 9.42 are decimals.