The value of `""_(216)log_(6) 39` is

Options

A. 117694
B. 117649
C. 216649
D. none of these

Answer

To find the value of \( 216^{\log_6 39} \), we can follow these steps:

### Step 1: Rewrite 216 in terms of base 6
We know that \( 216 = 6^3 \). Therefore, we can rewrite the expression as:
\[ 216^{\log_6 39} = (6^3)^{\log_6 39} \]

### Step 2: Apply the power of a power property
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \), we can simplify the expression:
\[ (6^3)^{\log_6 39} = 6^{3 \cdot \log_6 39} \]

### Step 3: Use the property of logarithms
We can use the property of logarithms that states \( a^{\log_a b} = b \). In our case, we can rewrite \( 3 \cdot \log_6 39 \) as:
\[ 6^{3 \cdot \log_6 39} = 39^3 \]

### Step 4: Calculate \( 39^3 \)
Now we need to calculate \( 39^3 \):
\[ 39^3 = 39 \times 39 \times 39 \]
Calculating this gives:
\[ 39 \times 39 = 1521 \]
Then,
\[ 1521 \times 39 = 59319 \]

### Final Answer
Thus, the value of \( 216^{\log_6 39} \) is:
\[ \boxed{59319} \]