Square Root of 3800

The square root is the inverse of the square of the number. 3800 is not a perfect square. The square root of 3800 is expressed in both radical and exponential form. In the radical form, it is expressed as √3800, whereas (3800)^(1/2) in the exponential form. √3800 ≈ 61.64414, 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 3800 is broken down into its prime factors

Step 1: Finding the prime factors of 3800 Breaking it down, we get 2 x 2 x 2 x 5 x 5 x 19: 2^3 x 5^2 x 19

Step 2: Now we found out the prime factors of 3800. The second step is to make pairs of those prime factors. Since 3800 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 3800 using prime factorization is impossible.

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 3800, we need to group it as 38 and 00.

Step 2: Now we need to find n whose square is 36. We can say n as ‘6’ because 6 × 6 is lesser than or equal to 38. Now the quotient is 6, after subtracting 36 from 38, the remainder is 2.

Step 3: Now let us bring down 00, 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 × n ≤ 200; let us consider n as 1, now 12 × 1 = 12.

Step 6: Subtract 12 from 200, the difference is 188, and the quotient is 61.

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 18800.

Step 8: Now we need to find the new divisor, which is 123 because 123 × 3 = 369.

Step 9: Subtracting 369 from 18800, we get the result 18431.

Step 10: Now the quotient is 61.3.

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 √3800 is 61.64.

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 3800 using the approximation method.

Step 1: Now we have to find the closest perfect square of √3800. The smallest perfect square of 3800 is 3600, and the largest perfect square of 3800 is 3844. √3800 falls somewhere between 60 and 62.

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 (3800 - 3600) ÷ (3844 - 3600) = 0.82. 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 + 0.82 = 60.82, so the square root of 3800 is approximately 60.82.

• 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, 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 a principal square root.
   
• Factors: Factors are the numbers you multiply together to get another number. For example, the factors of 6 are 1, 2, 3, and 6.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because 6 × 6 = 36.