Square Root of 3780

The square root is the inverse of the square of the number. 3780 is not a perfect square. The square root of 3780 is expressed in both radical and exponential form. In the radical form, it is expressed as √3780, whereas (3780)^(1/2) in the exponential form. √3780 ≈ 61.51478, 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 3780 is broken down into its prime factors.

Step 1: Finding the prime factors of 3780 Breaking it down, we get 2 x 2 x 3 x 3 x 5 x 7 x 9: 2^2 x 3^2 x 5 x 7 x 9

Step 2: Now we found out the prime factors of 3780. The second step is to make pairs of those prime factors. Since 3780 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating the square root of 3780 using prime factorization is not straightforward.

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 3780, we need to group it as 80 and 37.

Step 2: Now we need to find n whose square is 37. We can say n as ‘6’ because 6 x 6 = 36 is lesser than or equal to 37. Now the quotient is 6, and after subtracting 36 from 37, the remainder is 1.

Step 3: Now let us bring down 80, 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 ≤ 180. Let us consider n as 1, now 12 x 1 x 1 = 12.

Step 6: Subtract 12 from 180, the difference is 168, 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 16800.

Step 8: Now we need to find the new divisor that is 122 because 1221 x 1 = 1221.

Step 9: Subtracting 1221 from 16800 gives the result 4579.

Step 10: 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 √3780 is approximately 61.51.

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 3780 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √3780. The smallest perfect square less than 3780 is 3721, and the largest perfect square greater than 3780 is 3844. √3780 falls somewhere between 61 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 (3780 - 3721) ÷ (3844 - 3721) = 0.51478. 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 61 + 0.51478 = 61.51478, so the square root of 3780 is approximately 61.51478.

• 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, which 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 has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.
   
• Prime factorization: The process of expressing a number as the product of its prime factors.
   
• Long division method: A technique for finding the square root of a number by dividing and averaging, useful for non-perfect square numbers.