Find the rationalising factor of `5^(1/3)-2^(1/3)`.

Explanation

To find the rationalizing factor of \( 5^{1/3} - 2^{1/3} \), we can use the formula for the rationalizing factor of the difference of cube roots. The formula states:

\[ a^{1/3} - b^{1/3} = (a^{2/3} + a^{1/3}b^{1/3} + b^{2/3})(a^{1/3} - b^{1/3}) \]

### Step-by-Step Solution:

1. **Identify \( a \) and \( b \)**:
- Here, we let \( a = 5 \) and \( b = 2 \).

2. **Apply the formula**:
- According to the formula, the rationalizing factor of \( 5^{1/3} - 2^{1/3} \) is:
\[ 5^{2/3} + 5^{1/3} \cdot 2^{1/3} + 2^{2/3} \]

3. **Write the rationalizing factor**:
- Therefore, the rationalizing factor of \( 5^{1/3} - 2^{1/3} \) is:
\[ 5^{2/3} + 5^{1/3} \cdot 2^{1/3} + 2^{2/3} \]

4. **Verification**:
- To verify, we can multiply \( (5^{1/3} - 2^{1/3}) \) with \( (5^{2/3} + 5^{1/3} \cdot 2^{1/3} + 2^{2/3}) \):
\[ (5^{1/3} - 2^{1/3})(5^{2/3} + 5^{1/3} \cdot 2^{1/3} + 2^{2/3}) \]
- This should simplify to a rational number.

5. **Simplification**:
- Expanding the product:
\[ = 5^{1/3} \cdot 5^{2/3} + 5^{1/3} \cdot 5^{1/3} \cdot 2^{1/3} + 5^{1/3} \cdot 2^{2/3} - 2^{1/3} \cdot 5^{2/3} - 2^{1/3} \cdot 5^{1/3} \cdot 2^{1/3} - 2^{1/3} \cdot 2^{2/3} \]
- This simplifies to:
\[ 5^{1} - 2^{1} = 5 - 2 = 3 \]

Thus, the rationalizing factor of \( 5^{1/3} - 2^{1/3} \) is \( 5^{2/3} + 5^{1/3} \cdot 2^{1/3} + 2^{2/3} \).