To simplify the expression \((g - 12)^{-3}\), we can rewrite it in terms of positive exponents.
The expression can be rewritten as:
\[ (g - 12)^{-3} = \frac{1}{(g - 12)^3} \]
However, it seems there may be a slight misunderstanding because the options provided don't include this expression directly. Instead, if we're looking for a simplification directly in terms of \(g\), let’s analyze \( -3 \) in terms of \( g^n \):
When you have an expression like \( g^{-k} \), it means \( \frac{1}{g^k} \). We don’t see \( g \) specifically in this case, but we're simplifying this along the lines of the exponents.
If you meant to express \( g^{-3}\) instead of \( (g-12)^{-3}\):
\[ g^{-3} = \frac{1}{g^3} \]
Given your options, we can identify which one corresponds to the exponent:
- \( \frac{1}{g^{36}} \)
- \( \frac{1}{g^{9}} \)
- \( g^9 \)
- \( g^{36} \)
None of these options seem to directly tie back to \( (g - 12)^{-3} \). However, if we can assume \( g - 12 = g^{12}\) in some context, which is unlikely without the mathematical context, then the solutions don't seem to implicitly point to anything.
Can you please clarify if you were evaluating an exponent for \(g^k\) instead? If so, \( g^{-3}\) would be the right answer which isn't among the initial options clearly representing \( g \).
If it was indeed \(g^{-12}\), you would need to state more on that or seek further simplified forms if that was the approach you had.
Let me know how you would like to proceed!