A term of randomly chosen from the expansion of `(root(6)(4) + 1/(root(4)(5)))^(20)`. If the probability that it is a rational term is P, then 420P is euqal to

Answer

To solve the problem, we need to find the probability \( P \) that a randomly chosen term from the expansion of \( \left( \sqrt[6]{4} + \frac{1}{\sqrt[4]{5}} \right)^{20} \) is a rational term. Then, we will calculate \( 420P \).

### Step-by-step Solution:

1. **Rewrite the terms in the binomial expansion**:
\[ \sqrt[6]{4} = 2^{\frac{2}{6}} = 2^{\frac{1}{3}}, \quad \text{and} \quad \frac{1}{\sqrt[4]{5}} = 5^{-\frac{1}{4}}. \]
Thus, we can rewrite the expression as:
\[ \left( 2^{\frac{1}{3}} + 5^{-\frac{1}{4}} \right)^{20}. \]

2. **Identify the general term in the binomial expansion**:
The general term \( T_k \) in the expansion is given by:
\[ T_k = \binom{20}{k} \left(2^{\frac{1}{3}}\right)^{20-k} \left(5^{-\frac{1}{4}}\right)^{k} = \binom{20}{k} 2^{\frac{20-k}{3}} 5^{-\frac{k}{4}}. \]

3. **Determine conditions for rationality**:
For \( T_k \) to be rational, both exponents \( \frac{20-k}{3} \) and \( -\frac{k}{4} \) must be integers. This implies:
- \( 20 - k \) must be a multiple of 3.
- \( k \) must be a multiple of 4.

4. **Set up equations**:
Let \( 20 - k = 3m \) for some integer \( m \) (thus \( k = 20 - 3m \)).
Also, let \( k = 4n \) for some integer \( n \).

5. **Combine the equations**:
From \( k = 20 - 3m \) and \( k = 4n \), we have:
\[ 20 - 3m = 4n \implies 3m + 4n = 20. \]

6. **Find integer solutions**:
We can find non-negative integer solutions for \( m \) and \( n \):
- Rearranging gives \( 4n = 20 - 3m \).
- The right side must be non-negative, so \( 20 - 3m \geq 0 \) implies \( m \leq \frac{20}{3} \approx 6.67 \), hence \( m \) can take values \( 0, 1, 2, 3, 4, 5, 6 \).

Now, we can check each \( m \):
- \( m = 0 \): \( 4n = 20 \) → \( n = 5 \)
- \( m = 4 \): \( 4n = 8 \) → \( n = 2 \)
- \( m = 6 \): \( 4n = 2 \) → \( n = 0 \)

Thus, valid pairs \( (m, n) \) are \( (0, 5), (4, 2), (6, 0) \).

7. **Calculate valid \( k \) values**:
- For \( (0, 5) \): \( k = 20 - 3(0) = 20 \)
- For \( (4, 2) \): \( k = 20 - 3(4) = 8 \)
- For \( (6, 0) \): \( k = 20 - 3(6) = 2 \)

The valid \( k \) values are \( 20, 8, 2 \).

8. **Count the number of rational terms**:
There are 3 valid \( k \) values.

9. **Calculate total number of terms**:
The total number of terms in the expansion is \( 20 + 1 = 21 \).

10. **Calculate the probability \( P \)**:
\[ P = \frac{\text{Number of rational terms}}{\text{Total number of terms}} = \frac{3}{21} = \frac{1}{7}. \]

11. **Calculate \( 420P \)**:
\[ 420P = 420 \times \frac{1}{7} = 60. \]

### Final Answer:
Thus, \( 420P = 60 \).