Square Root of 3626

The square root is the inverse of the square of the number. 3626 is not a perfect square. The square root of 3626 is expressed in both radical and exponential form. In the radical form, it is expressed as √3626, whereas (3626)^(1/2) in the exponential form. √3626 ≈ 60.209, 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 3626 is broken down into its prime factors:

Step 1: Finding the prime factors of 3626 Breaking it down, we get 2 x 1813, where 1813 is a prime number. Since 3626 is not a perfect square, calculating √3626 using prime factorization is not straightforward as the digits cannot be grouped into pairs.

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 3626, we need to group it as 26 and 36.

Step 2: Now we need to find n whose square is less than or equal to 36. We can say n is 6 because 6^2 = 36. Now the quotient is 6 after subtracting 36 from 36, the remainder is 0.

Step 3: Bring down 26, which is the new dividend. Add the old divisor with the same number, 6 + 6, we get 12 as the new divisor.

Step 4: We need to find a number n such that 12n × n ≤ 2600. Let's try n = 2, then 122 × 2 = 244

Step 5: Subtract 244 from 260, and the difference is 16.

Step 6: Since the dividend is less than the divisor, we add a decimal point and bring down two zeroes, making the new dividend 1600.

Step 7: Now find the new divisor, 120, and repeat the steps until you achieve the desired precision.

So the square root of √3626 is approximately 60.209.

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 3626 using the approximation method.

Step 1: Find the closest perfect squares around 3626. The closest perfect square below 3626 is 3600 and above is 3721. √3626 falls between 60 and 61.

Step 2: Using the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) (3626 - 3600) / (3721 - 3600) ≈ 0.217 Add this decimal to 60: 60 + 0.217 ≈ 60.217

So the square root of 3626 is approximately 60.209 when calculated more precisely.

• Square root: A square root is the operation that finds the number which, when multiplied by itself, gives the original number. Example: 4² = 16, and the inverse operation, the square root, is √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written as a simple fraction p/q, where q is not zero and p and q are integers.
   
• Principal square root: The non-negative square root of a number. Both positive and negative roots exist, but the principal square root refers to the positive one.
   
• Prime factorization: The process of finding which prime numbers multiply together to make the original number.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing the number into pairs and solving step by step.