To evaluate \( (54)^{-3} \), we first need to rewrite the expression using the property of negative exponents:
\[ (54)^{-3} = \frac{1}{(54)^3} \]
Now we need to calculate \( (54)^3 \):
\[ 54^2 = 2916 \]
Next, calculate \( 54^3 = 54 \times 2916 \):
\[ 54^3 = 54 \times 2916 = 157464 \]
Now substituting back, we have:
\[ (54)^{-3} = \frac{1}{157464} \]
Thus, the value of \( (54)^{-3} \) is:
\[ \frac{1}{157464} \]
This value does not match any of the options listed, but we can simplify our approach using prime number factorization.
Rewriting 54 in its prime factorization:
\[ 54 = 2 \times 3^3 \]
Then raising it to the power of 3:
\[ 54^3 = (2 \times 3^3)^3 = 2^3 \times 3^9 = 8 \times 19683 = 157464 \]
Thus, we still confirm that:
\[ (54)^{-3} = \frac{1}{157464} \]
We ultimately verify once again that \( (54)^{-3} \) does not match the provided options.
If we had the options clarified, I could assist further in determining which fits. Based on standard simplification and computation, I’d conclude that none of the options accurately describe the evaluated expression \( (54)^{-3} = \frac{1}{157464} \).
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