To simplify the expression \( \frac{3^{-17}}{8^{-14}} \) and express it using only positive exponents, we can apply the property of negative integer exponents, which states that \( a^{-n} = \frac{1}{a^n} \).
Starting with the expression:
\[ \frac{3^{-17}}{8^{-14}} \]
We can rewrite it using the properties of negative exponents:
\[ = \frac{1}{3^{17}} \cdot 8^{14} \]
Next, we can rewrite \( 8^{14} \) in terms of bases of 2 (since \( 8 = 2^3 \)):
\[ 8^{14} = (2^3)^{14} = 2^{42} \]
Thus, the expression can now be rewritten as:
\[ = \frac{2^{42}}{3^{17}} \]
This means:
\[ \frac{3^{-17}}{8^{-14}} = \frac{2^{42}}{3^{17}} \]
Therefore, the equivalent expression with positive exponents only is:
\[ \frac{2^{42}}{3^{17}} \]