The value of `6^(log_10 40)*5^(log_10 36)` is

Options

A. 200
B. 216
C. 432
D. none of these

Explanation

To solve the problem \( 6^{\log_{10} 40} \cdot 5^{\log_{10} 36} \), we can follow these steps:

### Step 1: Set up the equation
Let \( x = 6^{\log_{10} 40} \cdot 5^{\log_{10} 36} \).

### Step 2: Take the logarithm of both sides
Taking the logarithm (base 10) of both sides gives us:
\[ \log_{10} x = \log_{10} (6^{\log_{10} 40}) + \log_{10} (5^{\log_{10} 36}) \]

### Step 3: Use the power rule of logarithms
Using the power rule of logarithms, we can simplify:
\[ \log_{10} x = \log_{10} 40 \cdot \log_{10} 6 + \log_{10} 36 \cdot \log_{10} 5 \]

### Step 4: Rewrite the logarithms of numbers
We can rewrite \( \log_{10} 40 \) and \( \log_{10} 36 \):
- \( \log_{10} 40 = \log_{10} (4 \cdot 10) = \log_{10} 4 + \log_{10} 10 = \log_{10} 4 + 1 \)
- \( \log_{10} 36 = \log_{10} (6^2) = 2 \log_{10} 6 \)

Substituting these back into the equation gives:
\[ \log_{10} x = (\log_{10} 4 + 1) \cdot \log_{10} 6 + 2 \log_{10} 6 \cdot \log_{10} 5 \]

### Step 5: Expand and simplify
Expanding the first term:
\[ \log_{10} x = \log_{10} 4 \cdot \log_{10} 6 + \log_{10} 6 + 2 \log_{10} 6 \cdot \log_{10} 5 \]
Combining like terms:
\[ \log_{10} x = \log_{10} 6 \cdot (\log_{10} 4 + 1 + 2 \log_{10} 5) \]

### Step 6: Further simplify
Recognizing that \( \log_{10} 4 = 2 \log_{10} 2 \):
\[ \log_{10} x = \log_{10} 6 \cdot (2 \log_{10} 2 + 1 + 2 \log_{10} 5) \]

### Step 7: Combine logarithmic terms
We can express \( 2 \log_{10} 2 + 2 \log_{10} 5 \) as \( 2(\log_{10} 2 + \log_{10} 5) = 2 \log_{10} 10 = 2 \):
\[ \log_{10} x = \log_{10} 6 \cdot (2 + 1) = 3 \log_{10} 6 \]

### Step 8: Exponentiate to solve for \( x \)
Exponentiating both sides gives:
\[ x = 6^3 \]
Calculating \( 6^3 \):
\[ x = 216 \]

### Final Answer
Thus, the value of \( 6^{\log_{10} 40} \cdot 5^{\log_{10} 36} \) is \( 216 \).

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