Square Root of 3888

The square root is the inverse of the square of the number. 3888 is not a perfect square. The square root of 3888 is expressed in both radical and exponential form. In the radical form, it is expressed as √3888, whereas (3888)^(1/2) in the exponential form. √3888 ≈ 62.349, 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 3888 is broken down into its prime factors.

Step 1: Finding the prime factors of 3888 Breaking it down, we get 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3: 2^4 × 3^5

Step 2: Now we found out the prime factors of 3888. The second step is to make pairs of those prime factors. Since 3888 is not a perfect square, therefore the digits of the number can’t be grouped in pairs completely.

Therefore, calculating 3888 using prime factorization alone 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 3888, we need to group it as 88 and 38.

Step 2: Now we need to find n whose square is closest to 38. We can say n is '6' because 6 × 6 = 36, which is less 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 88, making the new dividend 288. Add the old divisor with the same number (6 + 6) to get 12, which will be our new divisor.

Step 4: The new divisor will be 12n. We need to find the value of n where 12n × n ≤ 288. Considering n as 2, 12 × 2 = 24, and 24 × 2 = 48, which is incorrect. Trying n = 3, 12 × 3 = 36, 36 × 3 = 108, incorrect. For n = 4, 12 × 4 = 48, and 48 × 4 = 192, which is correct.

Step 5: Subtract 192 from 288 to get 96. 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 9600.

Step 6: Continue the process to find the next digits of the square root until you achieve the desired precision.

So, the square root of √3888 ≈ 62.349

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 3888 using the approximation method.

Step 1: Now we have to find the closest perfect square of √3888. The smallest perfect square less than 3888 is 3721 (61^2) and the largest perfect square more than 3888 is 3969 (63^2). √3888 falls somewhere between 61 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: (3888 - 3721) / (3969 - 3721) = 167 / 248 ≈ 0.673 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 + 0.673 ≈ 61.673, so the square root of 3888 is approximately 61.673.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. For example, 3^2 = 9 and the inverse of the square is the square root, which is √9 = 3.
   
• 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.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 4, 9, 16, etc., are perfect squares.
   
• Long division method: A method of dividing large numbers to find the quotient and remainder. It is also used to find the square root of non-perfect squares.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.