To calculate \( (54)^{-3} \), we first rewrite it using the property of negative exponents:
\[ (54)^{-3} = \frac{1}{(54)^3} \]
Next, we need to compute \( (54)^3 \):
\[ 54^3 = 54 \times 54 \times 54 \] Calculating this step-by-step:
- \( 54 \times 54 = 2916 \)
- Now, multiply that result by 54:
\[
2916 \times 54
\]
- First, we'll break it down: \[ 2916 \times 50 + 2916 \times 4 \]
- Calculate \( 2916 \times 50 \): \[ 2916 \times 50 = 145800 \]
- Now calculate \( 2916 \times 4 \): \[ 2916 \times 4 = 11664 \]
- Now, add these two results together: \[ 145800 + 11664 = 157464 \] Thus, \( 54^3 = 157464 \).
Putting it all together for \( (54)^{-3} \): \[ (54)^{-3} = \frac{1}{157464} \]
To find a simpler fraction, we can write this value as a decimal: \[ \frac{1}{157464} \approx 0.00000635 \] or in a fraction form, it remains \( \frac{1}{54^3} \) or simply \( \frac{1}{157464} \).
However, this result does not match the options you provided. The closest interpretation, if we also consider something like \( 64/125 \) or \( -64/125 \), could be incorrect options.
Since none of your provided options \( \text{−12564, 125 over 64, 64125, 64 over 125, 12564, 125 over 64, -64125} \) appear mathematically corresponding to \( (54)^{-3}\), please review any further or provide additional context for clarification.
If the actual question was to compute \( (54/64)^{-3} \) or similar, please clarify for exact computation.