Mario's error in simplifying the expression \((4 z^8)^{-3}\) is related to how he applied the negative exponent. The correct step when applying a negative exponent to a product is to apply the exponent to each factor separately.
The correct simplification of \((4 z^8)^{-3}\) is as follows:
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Apply the exponent \(-3\) to both the 4 and \(z^8\): \[ (4^{-3})(z^{8 \cdot -3}) = 4^{-3} z^{-24} \]
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This results in: \[ 4^{-3} = \frac{1}{4^3} = \frac{1}{64} \] and \(z^{-24} = \frac{1}{z^{24}}\).
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Combining both results gives: \[ (4 z^8)^{-3} = \frac{1}{64 z^{24}} \]
Therefore, the correct statement explaining Mario's error is: He should have applied the exponent –3 to both factors (4 and \(z^8\)) instead of just to \(z^8\), leading to \(4^{-3} z^{-24}\) instead of just \(4 z^{-24}\).
The closest option that reflects this reasoning is: He should have applied the exponent –3 to 4, and not to z, to get \(4^{-3} z^{8} = \frac{z^{8}}{64}\). This option, while slightly off in wording, captures the error in how he applied the exponent.