Find the cube root of 110592.

Answer

To find the cube root of 110592, we will follow these steps:

### Step 1: Prime Factorization
We will start by factoring the number 110592 into its prime factors.

1. **Divide by 2** (since 110592 is even):
- 110592 ÷ 2 = 55296
2. **Divide by 2** again:
- 55296 ÷ 2 = 27648
3. **Divide by 2** again:
- 27648 ÷ 2 = 13824
4. **Divide by 2** again:
- 13824 ÷ 2 = 6912
5. **Divide by 2** again:
- 6912 ÷ 2 = 3456
6. **Divide by 2** again:
- 3456 ÷ 2 = 1728
7. **Divide by 2** again:
- 1728 ÷ 2 = 864
8. **Divide by 2** again:
- 864 ÷ 2 = 432
9. **Divide by 2** again:
- 432 ÷ 2 = 216
10. **Divide by 2** again:
- 216 ÷ 2 = 108
11. **Divide by 2** again:
- 108 ÷ 2 = 54
12. **Divide by 2** again:
- 54 ÷ 2 = 27
13. **Now divide by 3** (since 27 is not divisible by 2):
- 27 ÷ 3 = 9
14. **Divide by 3** again:
- 9 ÷ 3 = 3
15. **Divide by 3** again:
- 3 ÷ 3 = 1

So, the complete prime factorization of 110592 is:
\[ 110592 = 2^{12} \times 3^3 \]

### Step 2: Grouping the Factors
To find the cube root, we need to group the prime factors into triplets.

- From \( 2^{12} \), we can form \( 4 \) groups of \( 2^3 \) (since \( 12 ÷ 3 = 4 \)).
- From \( 3^3 \), we can form \( 1 \) group of \( 3^3 \).

### Step 3: Taking the Cube Root
Now we can take the cube root of each group:

- From \( 2^{12} \), we take \( 2^4 \) (since \( 12 ÷ 3 = 4 \)).
- From \( 3^3 \), we take \( 3^1 \) (since \( 3 ÷ 3 = 1 \)).

Now, we multiply these results together:
\[ 2^4 \times 3^1 = 16 \times 3 = 48 \]

### Final Answer
Thus, the cube root of 110592 is:
\[ \sqrt[3]{110592} = 48 \]

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