Square Root of 3604

The square root is the inverse of the square of a number. 3604 is not a perfect square. The square root of 3604 is expressed in both radical and exponential form. In the radical form, it is expressed as √3604, whereas (3604)^(1/2) in the exponential form. √3604 ≈ 60.04, 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 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 3604 is broken down into its prime factors.

Step 1: Finding the prime factors of 3604 Breaking it down, we get 2 x 2 x 13 x 13 x 7: 2^2 x 13^2 x 7

Step 2: Now we found out the prime factors of 3604. Since 3604 is not a perfect square, the digits of the number can’t be grouped into complete pairs.

Therefore, calculating 3604 using prime factorization alone involves further steps.

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. For 3604, we group it as 04 and 36.

Step 2: Find n whose square is closest to 36. We can say n is ‘6’ because 6 x 6 = 36. The quotient is 6, and the remainder is 0 after subtracting 36 from 36.

Step 3: Bring down 04, which is the new dividend. Add the old divisor with the quotient, 6 + 6, to get 12, which will be our new divisor.

Step 4: The new divisor will be the sum of the current divisor and a number. We need to find n such that 12n × n ≤ 04.

Step 5: Since 12 x 0 x 0 = 0, subtract 0 from 04 to get a remainder of 4. The quotient is now 60.

Step 6: Since the dividend is less than the divisor, add a decimal point. Now the new dividend is 400.

Step 7: Find a new divisor such that 120n x n ≤ 400. If n = 3, then 120 x 3 x 3 = 360.

Step 8: Subtract 360 from 400 to get a remainder of 40. The quotient is 60.3.

Step 9: Continue until the desired precision is achieved.

The square root of √3604 ≈ 60.04.

The approximation method is another method for finding square roots, and it is an easy method to find the square root of a given number. Let us learn how to find the square root of 3604 using the approximation method.

Step 1: Find the closest perfect squares of √3604. The smallest perfect square less than 3604 is 3600, and the largest perfect square greater than 3604 is 3721. √3604 falls somewhere between 60 and 61.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Formula: (3604 - 3600) / (3721 - 3600) = 4 / 121 ≈ 0.033. Using the formula, we add the value we got initially to the decimal number: 60 + 0.033 ≈ 60.04.

So the square root of 3604 is approximately 60.04.

• 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 more commonly used in real-world applications, known as the principal square root.
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors. For example, prime factorization of 18 is 2 x 3 x 3.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing the number into pairs of digits starting from the decimal point.