Find the cube root of 262144 by prime factorisation method.

Answer

To find the cube root of 262144 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization
We need to factor 262144 into its prime factors. We will start dividing by the smallest prime number, which is 2.

1. **Divide by 2**:
- 262144 ÷ 2 = 131072
2. **Divide by 2**:
- 131072 ÷ 2 = 65536
3. **Divide by 2**:
- 65536 ÷ 2 = 32768
4. **Divide by 2**:
- 32768 ÷ 2 = 16384
5. **Divide by 2**:
- 16384 ÷ 2 = 8192
6. **Divide by 2**:
- 8192 ÷ 2 = 4096
7. **Divide by 2**:
- 4096 ÷ 2 = 2048
8. **Divide by 2**:
- 2048 ÷ 2 = 1024
9. **Divide by 2**:
- 1024 ÷ 2 = 512
10. **Divide by 2**:
- 512 ÷ 2 = 256
11. **Divide by 2**:
- 256 ÷ 2 = 128
12. **Divide by 2**:
- 128 ÷ 2 = 64
13. **Divide by 2**:
- 64 ÷ 2 = 32
14. **Divide by 2**:
- 32 ÷ 2 = 16
15. **Divide by 2**:
- 16 ÷ 2 = 8
16. **Divide by 2**:
- 8 ÷ 2 = 4
17. **Divide by 2**:
- 4 ÷ 2 = 2
18. **Divide by 2**:
- 2 ÷ 2 = 1

After dividing, we find that:
\[ 262144 = 2^{18} \]

### Step 2: Grouping the Factors
Next, we will group the prime factors into triplets since we are finding the cube root.

- From \( 2^{18} \), we can form triplets:
- \( (2 \times 2 \times 2) \) = 1 triplet
- We can form 6 triplets in total from \( 2^{18} \).

### Step 3: Finding the Cube Root
The cube root of a number can be found by taking one factor from each triplet. Since we have 6 triplets of 2, we can express the cube root as:
\[ \sqrt[3]{262144} = 2^{(18/3)} = 2^6 \]

Calculating \( 2^6 \):
\[ 2^6 = 64 \]

### Final Answer
Thus, the cube root of 262144 is:
\[ \boxed{64} \]

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