Find the value of `(y^(3))^(-2 log_(y)8)` is:

Options

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

Answer

To find the value of \((y^3)^{-2 \log_y 8}\), we can follow these steps:

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

### Step 2: Apply the power of a power property
Using the property of exponents, \((a^m)^n = a^{m \cdot n}\), we can rewrite the expression as:
\[ y^{3 \cdot (-2 \log_y 8)} \]
This simplifies to:
\[ y^{-6 \log_y 8} \]

### Step 3: Use the logarithmic identity
We can use the property of logarithms that states \(a^{\log_a b} = b\). Here, we have:
\[ y^{-6 \log_y 8} = 8^{-6} \]
This is because the base \(y\) in the exponent and the base of the logarithm are the same.

### Step 4: Calculate \(8^{-6}\)
Now we can compute \(8^{-6}\):
\[ 8^{-6} = \frac{1}{8^6} \]

### Step 5: Calculate \(8^6\)
Since \(8 = 2^3\), we can express \(8^6\) as:
\[ 8^6 = (2^3)^6 = 2^{18} \]

### Final Result
Thus, we have:
\[ 8^{-6} = \frac{1}{2^{18}} \]

Therefore, the value of \((y^3)^{-2 \log_y 8}\) is:
\[ \frac{1}{2^{18}} \]

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