Find the value of `(7^(3))^(-2log_(7)8)`

Options

A. A)`8^(-7)`
B. B)`6^(-8)`
C. C)`8^(-6)`
D. D)None of these

Answer

To solve the expression \((7^{3})^{-2\log_{7}8}\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (7^{3})^{-2\log_{7}8} \]

### 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:
\[ 7^{3 \cdot (-2\log_{7}8)} = 7^{-6\log_{7}8} \]

### Step 3: Use the logarithmic identity
We can apply the logarithmic identity \(a^{\log_{a}x} = x\) to rewrite the expression:
\[ 7^{-6\log_{7}8} = 8^{-6} \]

### Step 4: Final result
Thus, the value of the original expression is:
\[ \boxed{8^{-6}} \]

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