To find the cube root of 110592 by prime factorization, we need to break down the number into its prime factors. The prime factorization method involves finding the prime numbers that can be multiplied together to obtain the given number.

Prime factorization of 110592:
Step 1: Begin by dividing the number by the smallest prime number, which is 2.
110592 ÷ 2 = 55296

Step 2: Continue dividing the quotient by 2 until it is no longer divisible by 2.
55296 ÷ 2 = 27648
27648 ÷ 2 = 13824
13824 ÷ 2 = 6912
6912 ÷ 2 = 3456
3456 ÷ 2 = 1728
1728 ÷ 2 = 864
864 ÷ 2 = 432
432 ÷ 2 = 216
216 ÷ 2 = 108
108 ÷ 2 = 54
54 ÷ 2 = 27

Step 3: Now, the quotient obtained is an odd number, so we move on to the next prime number, which is 3.
27 ÷ 3 = 9

Step 4: Divide the obtained quotient by 3 until it is no longer divisible by 3.
9 ÷ 3 = 3

Step 5: The quotient obtained is a prime number, and we have completed the process.

Prime factorization of 110592:
110592 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3

Simplifying the prime factorization:
110592 = (2^12) × (3^6)

Now, to find the cube root of 110592, we need to find the cube root of each prime factor:
Cube root of (2^12) = 2^4 = 16
Cube root of (3^6) = 3^2 = 9

Therefore, the cube root of 110592 is:
∛110592 = 16 × 9 = 144

So, the cube root of 110592 is 144.