Square Root of 3900

The square root is the inverse of the square of the number. 3900 is not a perfect square. The square root of 3900 is expressed in both radical and exponential form. In the radical form, it is expressed as √3900, whereas (3900)^(1/2) in exponential form. √3900 ≈ 62.44998, 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 3900 is broken down into its prime factors.

Step 1: Finding the prime factors of 3900 Breaking it down, we get 2 x 2 x 3 x 5 x 5 x 13: 2² x 3¹ x 5² x 13¹

Step 2: Now we found the prime factors of 3900. The second step is to make pairs of those prime factors. Since 3900 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 3900 using prime factorization will yield an approximate value.

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, group the numbers from right to left. In the case of 3900, we need to group it as 39 and 00.

Step 2: Find a number whose square is less than or equal to 39. We can say n is ‘6’ because 6 x 6 = 36, which is less than 39. Now the quotient is 6 after subtracting 39 - 36, the remainder is 3.

Step 3: Bring down 00 to make the new dividend 300. Add the old divisor with the same number 6 + 6 to get 12, which will be our new divisor.

Step 4: Find a digit n such that 12n x n ≤ 300. Let n be 2. So, 122 x 2 = 244.

Step 5: Subtract the result from the dividend: 300 - 244 = 56.

Step 6: Since the dividend is less than the divisor, add a decimal point. Adding a decimal point allows us to add two zeroes to the dividend. The new dividend is 5600.

Step 7: Find the new divisor that is 124, because 1244 x 4 = 4976.

Step 8: Subtracting 4976 from 5600 gives the result 624.

Step 9: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.

So the square root of √3900 is approximately 62.45.

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 3900 using the approximation method.

Step 1: We have to find the closest perfect square to √3900. The smallest perfect square less than 3900 is 3600, and the largest perfect square more than 3900 is 4096. √3900 falls somewhere between 60 and 64.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using this formula: (3900 - 3600) / (4096 - 3600) = 300 / 496 ≈ 0.6048 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 60 + 0.6048 ≈ 60.60, so the square root of 3900 is approximately 62.45.

• Square root: A square root is the inverse of a square. Example: 4² = 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, the positive square root is often used due to its applications in the real world. This is known as the principal square root.
   
• Prime factorization: A method of expressing a number as a product of its prime factors. For example, the prime factorization of 3900 is 2² × 3¹ × 5² × 13¹.
   
• Long division method: A step-by-step method used to find the square root of non-perfect squares by dividing the number into smaller, more manageable parts.