Square Root of 4804

The square root is the inverse of the square of a number. 4804 is not a perfect square. The square root of 4804 is expressed in both radical and exponential forms. In the radical form, it is expressed as √4804, whereas (4804)^(1/2) in the exponential form. √4804 ≈ 69.327, 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 long-division and approximation methods 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 4804 is broken down into its prime factors.

Step 1: Finding the prime factors of 4804

Breaking it down, we get 2 x 2 x 7 x 7 x 49: 2^2 x 7^2 x 49

Step 2: Now that we found the prime factors of 4804, the second step is to make pairs of those prime factors. Since 4804 is not a perfect square, therefore the digits of the number can’t be grouped into pairs. Therefore, calculating 4804 using prime factorization is not straightforward.

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 4804, we need to group it as 04 and 48.

Step 2: Now we need to find n whose square is less than or equal to 48. We can say n is ‘6’ because 6 x 6 = 36, which is less than or equal to 48. Now the quotient is 6 after subtracting 48 - 36, the remainder is 12.

Step 3: Now let us bring down 04, which is the new dividend. Add the old divisor with the same number, 6 + 6, we get 12, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 12n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 12n x n ≤ 1204. Let us consider n as 9, now 129 x 9 = 1161.

Step 6: Subtract 1204 from 1161; the difference is 43, and the quotient is 69.

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 4300.

Step 8: Now we need to find the new divisor that is 693 because 693 x 6 = 4158.

Step 9: Subtracting 4158 from 4300, we get the result 142.

Step 10: Now the quotient is 69.3.

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 √4804 is approximately 69.33.

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 4804 using the approximation method.

Step 1: Now we have to find the closest perfect square of √4804. The smallest perfect square less than 4804 is 4761, and the largest perfect square greater than 4804 is 4900. √4804 falls somewhere between 69 and 70.

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 (4804 - 4761) ÷ (4900 - 4761) = 43 ÷ 139 ≈ 0.31 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 69 + 0.31 ≈ 69.31, so the square root of 4804 is approximately 69.31.

• 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 the reason it is also known as the principal square root.

• Prime factorization: Prime factorization is the process of writing a number as the product of its prime factors.

• Long division method: The long division method is a technique used to find the square root of a non-perfect square by iterative division and subtraction.