To rewrite the expression \( \frac{3^{17}}{8^{14}} \) with positive exponents only, we first recognize that we can apply the property of negative integer exponents, which states:
\[ a^{-n} = \frac{1}{a^n} \]
Here, we can rewrite \( 8^{14} \) in the denominator using negative exponents:
\[ \frac{3^{17}}{8^{14}} = 3^{17} \cdot 8^{-14} \]
Now, we also want to express \( 8 \) as a power of \( 2 \):
\[ 8 = 2^3 \]
Thus, we can rewrite \( 8^{14} \) as:
\[ 8^{14} = (2^3)^{14} = 2^{3 \cdot 14} = 2^{42} \]
Now substituting this into our earlier expression yields:
\[ 3^{17} \cdot 8^{-14} = 3^{17} \cdot (2^{42})^{-1} = 3^{17} \cdot 2^{-42} \]
At this point, the expression is:
\[ 3^{17} \cdot 2^{-42} \]
It’s helpful to write \( 2^{-42} \) in the numerator:
\[ 3^{17} \cdot 2^{-42} = \frac{3^{17}}{2^{42}} \]
Thus, we have:
\[ \frac{3^{17}}{8^{14}} = 3^{17} \cdot 2^{-42} \]
And since we started with negative exponents, we now have the final expression:
\[ \frac{3^{17}}{2^{42}} \]
This results in an expression equivalent to \( \frac{3^{17}}{8^{14}} \) with positive exponents only as:
\[ 3^{17} \cdot 2^{-42} \]
So, the final result can also be written as:
\[ \frac{3^{17}}{2^{42}} \]
or:
\[ 3^{17} \cdot 2^{-42} \]
This completes the transformation!