Square Root of 804

The square root is the inverse of the square of the number. 804 is not a perfect square. The square root of 804 is expressed in both radical and exponential form. In the radical form, it is expressed as √804, whereas (804)^(1/2) in the exponential form. √804 ≈ 28.358, 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 804 is broken down into its prime factors:

Step 1: Finding the prime factors of 804 Breaking it down, we get 2 x 2 x 3 x 67: 2^2 x 3^1 x 67^1

Step 2: Now we found out the prime factors of 804. The second step is to make pairs of those prime factors. Since 804 is not a perfect square, therefore, the digits of the number can’t be grouped in pairs.

Therefore, calculating 804 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 804, we need to group it as 04 and 80.

Step 2: Now we need to find n whose square is 80. We can say n as ‘8’ because 8 × 8 is equal to 64, which is lesser than 80. Now the quotient is 8, and after subtracting 64 from 80, the remainder is 16.

Step 3: Now let us bring down 04, which is the new dividend. Add the old divisor with the same number 8 + 8, we get 16, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 16n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 16n × n ≤ 1604, let us consider n as 9, now 16 x 9 = 144.

Step 6: The difference is 1604 - 144 = 1460, and 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. Now the new dividend is 146000.

Step 8: Now we need to find the new divisor that is 283 because 283 x 3 = 849.

Step 9: Subtracting 849 from 1460, we get the result 611.

Step 10: Now the quotient is 28.3

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 √804 ≈ 28.36

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 804 using the approximation method.

Step 1: Now we have to find the closest perfect square of √804.

The smallest perfect square less than 804 is 784, and the largest perfect square greater than 804 is 841. √804 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 (804 - 784) ÷ (841 - 784) ≈ 0.35.

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 + 0.35 = 28.35, so the square root of 804 is approximately 28.35.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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 has more prominence due to its uses in the real world. That is why it is also known as the principal square root.

• Prime factorization: Breaking down a number into its basic prime number factors. Example: 804 = 2 x 2 x 3 x 67.

• Approximation: A method of finding a number close to the exact value, often used with irrational numbers.