To rewrite the expression \(\frac{3^{17}}{8^{14}}\) with positive exponents only, we need to apply the property of negative integer exponents, which states that \(a^{-n} = \frac{1}{a^n}\).
First, we can express the denominator \(8^{14}\) with a negative exponent:
\[ \frac{3^{17}}{8^{14}} = 3^{17} \cdot 8^{-14} \]
Now our expression is:
\[ 3^{17} \cdot 8^{-14} \]
To further convert this to only positive exponents, we keep \(3^{17}\) as it is and can express \(8^{-14}\) in terms of positive exponent by writing:
\[ 3^{17} \cdot \frac{1}{8^{14}} = 3^{17} \cdot 8^{-14} \]
However, to express it in a single term with positive exponent, we write it as:
\[ \frac{3^{17}}{8^{14}} \]
Now, we need to ensure all bases have positive exponents only.
Since \(8 = 2^3\), we can rewrite \(8^{14}\) as:
\[ 8^{14} = (2^3)^{14} = 2^{42} \]
Thus, our expression now becomes:
\[ \frac{3^{17}}{2^{42}} \]
To write this using positive exponents, we can maintain this formatting. The final expression equivalent to \(\frac{3^{17}}{8^{14}}\) with positive exponents only is:
\[ \frac{3^{17}}{2^{42}} \]
Alternatively, if you want to write it in a single fraction form with both terms in the numerator, it can be expressed as:
\[ \frac{3^{17}}{2^{42}} = 3^{17} \cdot 2^{-42} \]
In summary, the expression \(\frac{3^{17}}{8^{14}}\) can be expressed with positive exponents as:
\[ 3^{17} \cdot 2^{-42} \]
Or alternatively, keep it in the positive form as:
\[ \frac{3^{17}}{2^{42}} \]