The reciprocal of `2/(log_(4)(2000)^(6))+3/(log_(5)(2000)^(6))` is _______.

Explanation

To find the reciprocal of the expression \( \frac{2}{\log_{4}(2000)^{6}} + \frac{3}{\log_{5}(2000)^{6}} \), we can follow these steps:

### Step 1: Rewrite the logarithmic expressions
Let \( x = \log_{2000}^{6} \). Then we can rewrite the expression as:
\[ \frac{2}{\log_{4}(x)} + \frac{3}{\log_{5}(x)} \]

### Step 2: Use the change of base formula
Using the change of base formula for logarithms, we have:
\[ \log_{4}(x) = \frac{\log(x)}{\log(4)} \quad \text{and} \quad \log_{5}(x) = \frac{\log(x)}{\log(5)} \]
Substituting these into the expression gives:
\[ \frac{2 \log(4)}{\log(x)} + \frac{3 \log(5)}{\log(x)} \]

### Step 3: Combine the fractions
Since both terms have a common denominator, we can combine them:
\[ \frac{2 \log(4) + 3 \log(5)}{\log(x)} \]

### Step 4: Factor out \( \frac{1}{\log(x)} \)
Now we can express the entire expression as:
\[ \frac{1}{\log(x)} (2 \log(4) + 3 \log(5)) \]

### Step 5: Simplify using logarithmic properties
Using the property \( n \log(m) = \log(m^n) \):
\[ 2 \log(4) = \log(4^2) = \log(16) \quad \text{and} \quad 3 \log(5) = \log(5^3) = \log(125) \]
Thus, we can rewrite the expression as:
\[ \frac{1}{\log(x)} (\log(16) + \log(125)) \]

### Step 6: Combine the logarithms
Using the property \( \log(a) + \log(b) = \log(ab) \):
\[ \log(16) + \log(125) = \log(16 \times 125) = \log(2000) \]
So, we have:
\[ \frac{\log(2000)}{\log(x)} \]

### Step 7: Substitute back for \( x \)
Recall that \( x = \log_{2000}^{6} \), so:
\[ \log(x) = \log(\log_{2000}^{6}) = 6 \log(2000) \]
Thus, the expression simplifies to:
\[ \frac{\log(2000)}{6 \log(2000)} = \frac{1}{6} \]

### Step 8: Find the reciprocal
The reciprocal of \( \frac{1}{6} \) is:
\[ 6 \]

### Final Answer
The reciprocal of \( \frac{2}{\log_{4}(2000)^{6}} + \frac{3}{\log_{5}(2000)^{6}} \) is \( \boxed{6} \).