The cube root of 10648 is

Answer

To find the cube root of 10648 using the prime factorization method, we can follow these steps:

### Step-by-Step Solution:

1. **Find the Prime Factorization of 10648:**
- Start dividing 10648 by the smallest prime number, which is 2.
- 10648 ÷ 2 = 5324
- 5324 ÷ 2 = 2662
- 2662 ÷ 2 = 1331
- 1331 is not divisible by 2, so we move to the next prime number, which is 3.
- 1331 ÷ 3 = not divisible, so we try 5.
- 1331 ÷ 5 = not divisible, so we try 7.
- 1331 ÷ 7 = not divisible, so we try 11.
- 1331 ÷ 11 = 121
- 121 ÷ 11 = 11
- 11 ÷ 11 = 1

Therefore, the prime factorization of 10648 is:
\[ 10648 = 2^3 \times 11^3 \]

2. **Apply the Cube Root:**
- We need to find the cube root of 10648, which can be expressed as:
\[ \sqrt[3]{10648} = \sqrt[3]{2^3 \times 11^3} \]

3. **Simplify the Cube Root:**
- The cube root of a product is the product of the cube roots:
\[ \sqrt[3]{2^3} \times \sqrt[3]{11^3} = 2 \times 11 \]

4. **Calculate the Final Result:**
- Now, multiply the results:
\[ 2 \times 11 = 22 \]

Thus, the cube root of 10648 is **22**.