In the binomial expansion of `( root(3) (4) + sqrt(2) )^5` find the term which does not contain Irrational expression.

Answer

To solve the problem, we need to find the term in the binomial expansion of \( \left( \sqrt[3]{4} + \sqrt{2} \right)^5 \) that does not contain any irrational expression.

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term \( T_{r+1} \) in the binomial expansion of \( (x + y)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} x^{n-r} y^r \]
Here, \( n = 5 \), \( x = \sqrt[3]{4} \), and \( y = \sqrt{2} \).

2. **Write the General Term for Our Expression**:
For our case, the general term becomes:
\[ T_{r+1} = \binom{5}{r} \left( \sqrt[3]{4} \right)^{5-r} \left( \sqrt{2} \right)^r \]

3. **Simplify the General Term**:
We can rewrite \( \sqrt[3]{4} \) as \( 4^{1/3} \) and \( \sqrt{2} \) as \( 2^{1/2} \):
\[ T_{r+1} = \binom{5}{r} \cdot 4^{(5-r)/3} \cdot 2^{r/2} \]

4. **Determine Conditions for Rationality**:
For the term to be rational, both \( 4^{(5-r)/3} \) and \( 2^{r/2} \) must be rational numbers. This requires:
- \( \frac{5-r}{3} \) must be an integer (i.e., \( 5 - r \) must be a multiple of 3).
- \( \frac{r}{2} \) must also be an integer (i.e., \( r \) must be even).

5. **Find Possible Values of \( r \)**:
Since \( r \) must be even, let \( r = 2k \) where \( k \) is an integer. The possible values of \( r \) from 0 to 5 are 0, 2, and 4.
- For \( r = 0 \): \( 5 - r = 5 \) (not a multiple of 3)
- For \( r = 2 \): \( 5 - r = 3 \) (is a multiple of 3)
- For \( r = 4 \): \( 5 - r = 1 \) (not a multiple of 3)

Thus, the only valid value for \( r \) is 2.

6. **Calculate the Specific Term**:
Substitute \( r = 2 \) into the general term:
\[ T_{3} = \binom{5}{2} \left( \sqrt[3]{4} \right)^{5-2} \left( \sqrt{2} \right)^2 \]
\[ = \binom{5}{2} \left( \sqrt[3]{4} \right)^{3} \cdot 2 \]
\[ = \binom{5}{2} \cdot 4 \cdot 2 \]
\[ = 10 \cdot 4 \cdot 2 = 80 \]

### Conclusion:
The term in the binomial expansion of \( \left( \sqrt[3]{4} + \sqrt{2} \right)^5 \) that does not contain any irrational expression is \( 80 \).