To rewrite \( 8^{2 - \sqrt{3}} \), we can use the properties of exponents.
We can express \( 8 \) as \( 2^3 \):
\[ 8 = 2^3 \]
Now substituting this into the original expression:
\[ 8^{2 - \sqrt{3}} = (2^3)^{2 - \sqrt{3}} \]
Using the power of a power property \((a^m)^n = a^{m \cdot n}\), we get:
\[ (2^3)^{2 - \sqrt{3}} = 2^{3(2 - \sqrt{3})} \]
Now simplifying the exponent:
\[ 3(2 - \sqrt{3}) = 6 - 3\sqrt{3} \]
Thus, we can rewrite the expression as:
\[ 8^{2 - \sqrt{3}} = 2^{6 - 3\sqrt{3}} \]
The rational number that could be used as an exponent is \( 6 - 3\sqrt{3} \).
Therefore, the simplified expression is:
\[ 8^{2 - \sqrt{3}} = 2^{6 - 3\sqrt{3}} \]