The solution of the equation `2^(3//log_(3)x)=1//64` is

Options

A. `3`
B. `(1)/(3)`
C. `(1)/(sqrt(3))`
D. none

Answer

To solve the equation \( \frac{2^3}{\log_3 x} = \frac{1}{64} \), we will follow these steps:

### Step 1: Rewrite \( \frac{1}{64} \)
We know that \( 64 = 2^6 \), so we can rewrite \( \frac{1}{64} \) as:
\[ \frac{1}{64} = 2^{-6} \]
Thus, our equation becomes:
\[ \frac{2^3}{\log_3 x} = 2^{-6} \]

### Step 2: Set the bases equal
Since both sides of the equation are powers of 2, we can equate the exponents:
\[ \frac{3}{\log_3 x} = -6 \]

### Step 3: Cross-multiply
To eliminate the fraction, we cross-multiply:
\[ 3 = -6 \log_3 x \]

### Step 4: Solve for \( \log_3 x \)
Now, divide both sides by -6:
\[ \log_3 x = -\frac{3}{6} = -\frac{1}{2} \]

### Step 5: Convert from logarithmic to exponential form
Using the definition of logarithms, we can convert this to exponential form:
\[ x = 3^{-\frac{1}{2}} \]

### Step 6: Simplify the expression
This can be simplified further:
\[ x = \frac{1}{\sqrt{3}} \]

### Final Answer
Thus, the solution to the equation is:
\[ x = \frac{1}{\sqrt{3}} \]