Square Root of 806

The square root is the inverse of the square of the number. 806 is not a perfect square. The square root of 806 is expressed in both radical and exponential form. In the radical form, it is expressed as √806, whereas (806)^(1/2) is the exponential form. √806 ≈ 28.403, 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 806 is broken down into its prime factors.

Step 1: Finding the prime factors of 806 Breaking it down, we get 2 x 13 x 31: 2^1 x 13^1 x 31^1

Step 2: Now we found out the prime factors of 806. The second step is to make pairs of those prime factors. Since 806 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 806 using prime factorization is impractical.

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 806, we need to group it as 06 and 8.

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

Step 3: Now let us bring down 06, which is the new dividend. Add the old divisor with the same number 2 + 2; we get 4, which will be our new divisor.

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

Step 5: The next step is finding 4n x n ≤ 406. Let us consider n as 7, now 47 x 7 = 329.

Step 6: Subtract 406 from 329; the difference is 77, and the quotient is 27.

Step 7: Since the dividend is less than the divisor, we add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 7700.

Step 8: Now we need to find the new divisor, which is 549 because 549 x 9 = 4941.

Step 9: Subtracting 4941 from 7700, we get the result 2759.

Step 10: Now the quotient is 28.4

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 √806 is approximately 28.4.

The approximation method is another method for finding the 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 806 using the approximation method.

Step 1: Now we have to find the closest perfect square of √806.

The smallest perfect square less than 806 is 784, and the largest perfect square greater than 806 is 841. √806 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 (806 - 784) ÷ (841 - 784) = 0.386

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 + 0.386 ≈ 28.39, so the square root of 806 is approximately 28.39.

• Square Root: A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, since 4 x 4 = 16.

• Irrational Number: An irrational number cannot be expressed as a fraction p/q where p and q are integers and q ≠ 0. Examples include √2 and √3.

• Perfect Square: A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because it is 6 x 6.

• Long Division Method: A method used to find the square root of non-perfect squares by dividing the number into groups and repeatedly finding remainders.

• Prime Factorization: The process of expressing a number as a product of its prime factors. This method is useful for simplifying roots and understanding number properties.