Find the square root of 7744 by prime factorisation method.

To find the square root of 7744 using the prime factorization method, we first need to break down the number into its prime factors. This method is effective because the square root of a number can be easily determined once we have its prime factorization.

Step-by-Step Prime Factorization

Let's start by dividing 7744 by the smallest prime number, which is 2, and continue dividing until we can no longer divide evenly.

• 7744 ÷ 2 = 3872
• 3872 ÷ 2 = 1936
• 1936 ÷ 2 = 968
• 968 ÷ 2 = 484
• 484 ÷ 2 = 242
• 242 ÷ 2 = 121

At this point, we can no longer divide by 2 since 121 is not even. Next, we check the next prime number, which is 3. Since 121 is not divisible by 3, we move to the next prime number, which is 5. Again, 121 is not divisible by 5. The next prime number is 7, and 121 is not divisible by 7 either. Finally, we try 11:

• 121 ÷ 11 = 11
• 11 ÷ 11 = 1

Compiling the Prime Factors

Now that we have completed the division, we can compile the prime factors we found:

• 2 appears 6 times (2^6)
• 11 appears 2 times (11^2)

Thus, the prime factorization of 7744 can be expressed as:

7744 = 2^6 × 11^2

Finding the Square Root

To find the square root, we take the square root of each prime factor raised to its respective power. The square root of a number raised to an even power is simply the base raised to half of that power:

• √(2^6) = 2^(6/2) = 2^3 = 8
• √(11^2) = 11^(2/2) = 11^1 = 11

Now, we multiply these results together to find the square root of 7744:

√7744 = 8 × 11 = 88

Final Result

Therefore, the square root of 7744 is 88. This method not only helps in finding the square root but also reinforces the understanding of prime factorization and its applications in mathematics.