Square Root of 3856

The square root is the inverse of the square of a number. 3856 is not a perfect square. The square root of 3856 is expressed in both radical and exponential forms. In the radical form, it is expressed as √3856, whereas (3856)^(1/2) in the exponential form. √3856 ≈ 62.060, 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 3856 is broken down into its prime factors.

Step 1: Finding the prime factors of 3856 Breaking it down, we get 2 x 2 x 2 x 2 x 241: 2^4 x 241

Step 2: Now we found out the prime factors of 3856. The second step is to make pairs of those prime factors. Since 3856 is not a perfect square, therefore the digits of the number can’t be grouped into pairs.

Therefore, calculating 3856 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 3856, we need to group it as 56 and 38.

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

Step 3: Now let us bring down 56, 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 x n ≤ 256. Let us consider n as 2, now 12 x 2 x 2 = 48.

Step 6: Subtract 256 from 48, the difference is 208, and the quotient is 62.

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 20800.

Step 8: Now we need to find the new divisor, which is 124 because 1249 x 9 = 11241.

Step 9: Subtracting 11241 from 20800, we get the result 9569.

Step 10: Now the quotient is 62.0.

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 √3856 is approximately 62.06.

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 3856 using the approximation method.

Step 1: Now we have to find the closest perfect square of √3856. The smallest perfect square less than 3856 is 3844, and the largest perfect square greater than 3856 is 3969. √3856 falls somewhere between 62 and 63.

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 (3856 - 3844) ÷ (3969 - 3844) = 12/125 = 0.096. 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 62 + 0.096 = 62.096, so the square root of 3856 is approximately 62.096.

• 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 has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.
   
• Long division method: A method used to find the square roots of non-perfect squares by dividing the number and finding approximate values.
   
• Approximation method: A method used to estimate the square roots of numbers by identifying two close perfect squares and using them to find an approximate value.