2^17x3^8/6^8x8^3

To simplify the expression \( \frac{2^{17} \cdot 3^8}{6^8 \cdot 8^{32}} \cdot \frac{3^8}{6^8 \cdot 8^3} \), we will first break down the terms in the denominator.

Recall that:
- \( 6 = 2 \cdot 3 \)
- \( 8 = 2^3 \)

Let's start simplifying the denominators.

### Simplifying \( 6^8 \):
\[
6^8 = (2 \cdot 3)^8 = 2^8 \cdot 3^8
\]

### Simplifying \( 8^{32} \):
\[
8^{32} = (2^3)^{32} = 2^{96}
\]

### Now substituting back into the expression:
The full expression becomes:
\[
\frac{2^{17} \cdot 3^8}{(2^8 \cdot 3^8) \cdot 2^{96}} \cdot \frac{3^8}{(2^8 \cdot 3^8) \cdot 8^3}
\]

Now we substitute \( 8^3 = (2^3)^3 = 2^9 \):
\[
\frac{3^8}{(2^8 \cdot 3^8) \cdot 2^9} = \frac{3^8}{2^{8+9} \cdot 3^8} = \frac{3^8}{2^{17} \cdot 3^8} = \frac{1}{2^{17}}
\]

### Rewriting the full expression:
Now the expression simplifies to:
\[
\frac{2^{17} \cdot 3^8}{(2^8 \cdot 3^8) \cdot 2^{96}} \cdot \frac{1}{2^{17}}
\]

### Combining the denominators:
Now, combine the \( 2^{8} \) and \( 2^{96} \):
\[
= \frac{2^{17} \cdot 3^8}{2^{8 + 96} \cdot 3^8} \cdot \frac{1}{2^{17}} = \frac{2^{17} \cdot 3^8}{2^{104} \cdot 3^8} \cdot \frac{1}{2^{17}}
\]

### Canceling \( 3^8 \):
The \( 3^8 \) in the numerator and denominator cancel out:
\[
= \frac{2^{17}}{2^{104}} \cdot \frac{1}{2^{17}} = \frac{1}{2^{104 - 17}} = \frac{1}{2^{87}}
\]

Thus, the final simplified expression is:
\[
\frac{1}{2^{87}}
\]