Square Root of 11664

The square root is the inverse of the square of the number. 11664 is a perfect square. The square root of 11664 can be expressed in both radical and exponential form. In the radical form, it is expressed as √11664, whereas in the exponential form it is expressed as (11664)^(1/2). √11664 = 108, which is a rational number because it can 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. For non-perfect square numbers, the long-division method and approximation method are often 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 11664 is broken down into its prime factors.

Step 1: Finding the prime factors of 11664.

Breaking it down, we get 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 3: 2^4 x 3^6

Step 2: Now we found out the prime factors of 11664. The second step is to make pairs of those prime factors. Since 11664 is a perfect square, the digits of the number can be grouped in pairs. Therefore, calculating √11664 using prime factorization is possible.

Step 3: Pairing the factors, we get (2^2) x (3^3). Taking one factor from each pair, we find the square root: 2 x 3 x 3 = 108.

The long division method is particularly useful for non-perfect square numbers. However, it can also be applied to perfect squares. 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 11664, we need to group it as 64 and 116.

Step 2: Now we need to find n whose square is closest to the first group. We can say n as ‘10’ because 10 x 10 is 100, which is lesser than or equal to 116. Now the quotient is 10, and after subtracting 100 from 116, the remainder is 16.

Step 3: Now let us bring down 64, which is the new dividend. Add the old divisor with the same number, 10 + 10, we get 20, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 20n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 20n × n ≤ 1664. Let us consider n as 8, now 208 x 8 = 1664.

Step 6: Subtracting 1664 from 1664 gives the remainder as 0.

Step 7: The quotient is 108, and since the remainder is zero, the process ends here.

So, the square root of √11664 is 108.

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. However, since 11664 is a perfect square, the approximation method is not necessary in this case.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. For example, 108 x 108 = 11664, thus √11664 = 108.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 11664 is a perfect square because it equals 108 squared.
   
• Rational number: A rational number can be expressed as a fraction with integer values in the numerator and a non-zero integer in the denominator. For example, 108 is a rational number.
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors. For example, the prime factorization of 11664 is 2^4 x 3^6.
   
• Factor: A factor is a number that divides another number without leaving a remainder. For example, 108 is a factor of 11664.