The value of `(log_(10)2)^(3)+log_(10)8 * log_(10) 5 + (log_(10)5)^(3)` is _______.

Explanation

To solve the expression \((\log_{10} 2)^3 + \log_{10} 8 \cdot \log_{10} 5 + (\log_{10} 5)^3\), we will follow these steps:

### Step 1: Define Variables
Let:
- \( p = \log_{10} 2 \)
- \( q = \log_{10} 5 \)

### Step 2: Rewrite the Expression
The expression can be rewritten using \( p \) and \( q \):
\[ p^3 + \log_{10} 8 \cdot q + q^3 \]

### Step 3: Simplify \(\log_{10} 8\)
We know that:
\[ \log_{10} 8 = \log_{10} (2^3) = 3 \log_{10} 2 = 3p \]
So, we can substitute this into the expression:
\[ p^3 + 3p \cdot q + q^3 \]

### Step 4: Use the Identity for Cubes
We can use the identity:
\[ p^3 + q^3 = (p + q)(p^2 - pq + q^2) \]
Thus, we can rewrite our expression as:
\[ (p^3 + q^3) + 3pq = (p + q)(p^2 - pq + q^2) + 3pq \]

### Step 5: Find \( p + q \)
Using the property of logarithms:
\[ p + q = \log_{10} 2 + \log_{10} 5 = \log_{10} (2 \cdot 5) = \log_{10} 10 = 1 \]

### Step 6: Substitute \( p + q \)
Now substituting \( p + q = 1 \) into our expression:
\[ p^3 + q^3 + 3pq = (1)(p^2 - pq + q^2) + 3pq \]

### Step 7: Calculate \( p^2 + q^2 \)
We know:
\[ p^2 + q^2 = (p + q)^2 - 2pq = 1^2 - 2pq = 1 - 2pq \]

### Step 8: Substitute Back
Now substituting \( p^2 + q^2 \):
\[ p^3 + q^3 + 3pq = (1)(1 - 2pq) + 3pq = 1 - 2pq + 3pq = 1 + pq \]

### Step 9: Final Expression
Thus, we have:
\[ p^3 + q^3 + 3pq = 1 + pq \]

### Step 10: Find \( pq \)
Calculating \( pq \):
\[ pq = \log_{10} 2 \cdot \log_{10} 5 \]
Using the logarithmic property:
\[ pq = \log_{10} (2^x) \cdot \log_{10} (5^y) \text{ (not needed for final answer)} \]

However, we don't need the exact value of \( pq \) to find the final answer, as we can see from the earlier steps that the expression simplifies to \( 1 + pq \).

### Final Answer
The value of the expression is:
\[ \boxed{1} \]