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 identity for the difference of cube roots. The rationalizing factor can be derived from the following identity:

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

Where \( a = 5 \) and \( b = 2 \).

### Step-by-Step Solution:

1. **Identify \( a \) and \( b \)**:
- Let \( a = 5 \) and \( b = 2 \).

2. **Apply the identity**:
- The rationalizing factor of \( 5^{1/3} - 2^{1/3} \) is given by:
\[ 5^{2/3} + 5^{1/3} \cdot 2^{1/3} + 2^{2/3} \]

3. **Calculate each term**:
- \( 5^{2/3} \) is the cube root of \( 5^2 \), which is \( \sqrt[3]{25} \).
- \( 2^{2/3} \) is the cube root of \( 2^2 \), which is \( \sqrt[3]{4} \).
- \( 5^{1/3} \cdot 2^{1/3} \) can be expressed as \( (5 \cdot 2)^{1/3} = 10^{1/3} \).

4. **Combine the terms**:
- Therefore, the rationalizing factor can be expressed as:
\[ 5^{2/3} + 10^{1/3} + 2^{2/3} \]

5. **Final expression**:
- Thus, the rationalizing factor of \( 5^{1/3} - 2^{1/3} \) is:
\[ 5^{2/3} + 10^{1/3} + 2^{2/3} \]