Square Root of 5904

The square root is the inverse of the square of the number. 5904 is not a perfect square. The square root of 5904 is expressed in both radical and exponential form. In the radical form, it is expressed as √5904, whereas (5904)^(1/2) in the exponential form. √5904 ≈ 76.849, 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, methods like 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 5904 is broken down into its prime factors:

Step 1: Finding the prime factors of 5904 Breaking it down, we get 2 x 2 x 2 x 2 x 3 x 3 x 5 x 41: 2^4 x 3^2 x 5^1 x 41^1

Step 2: Now we have found the prime factors of 5904. The second step is to make pairs of those prime factors. Since 5904 is not a perfect square, the digits of the number can’t be grouped into complete pairs.

Therefore, calculating 5904 using prime factorization as a perfect square 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: Group the numbers from right to left. In the case of 5904, we need to group it as 04 and 59.

Step 2: Find a number whose square is less than or equal to 59. We can say this number is ‘7’ because 7 x 7 = 49, which is less than 59. Now the quotient is 7 and the remainder is 10.

Step 3: Bring down 04, making the new dividend 1004.

Step 4: Double the quotient and place it as the new divisor's first digit, i.e., 14_.

Step 5: Now find a digit to fill in the blank such that 14_ times that digit is less than or equal to 1004. The suitable digit is 7 as 147 x 7 = 1029, which is too high, so try 146 x 6 = 876.

Step 6: Subtract 876 from 1004, the remainder is 128.

Step 7: Since the remainder is less than the divisor, add a decimal point and bring down pairs of zeros.

Step 8: Continue this process to find more decimal places.

So, the square root of √5904 ≈ 76.849

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 5904 using the approximation method.

Step 1: Find the closest perfect squares of √5904.

The smallest perfect square less than 5904 is 5776 and the largest perfect square greater than 5904 is 5929. √5904 falls somewhere between 76 and 77.

Step 2: Apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula: (5904 - 5776) ÷ (5929 - 5776) = 128 ÷ 153 = 0.8366

Using the formula, we identified the decimal point of our square root. The next step is adding the integer part, which is 76, so 76 + 0.8366 ≈ 76.84.

Therefore, the square root of 5904 is approximately 76.84.

• 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 is commonly used in real-world applications, known as the principal square root.

• Prime factorization: Prime factorization is expressing a number as a product of its prime numbers. For example, the prime factorization of 20 is 2^2 × 5.

• Decimal: If a number has a whole number and a fraction in a single number, it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.