To find which expression simplifies to \(\frac{1}{256} t^{28}\), we will simplify each of the provided expressions one by one.
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Expression: \((256 t^{-7})^4\) \[ = 256^4 (t^{-7})^4 = 256^4 t^{-28} \] Since \(256 = 2^8\), we have \(256^4 = (2^8)^4 = 2^{32}\), so: \[ = 2^{32} t^{-28} \] This is not equivalent to \(\frac{1}{256} t^{28}\).
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Expression: \((4 t^{28})^{-4}\) \[ = (4^{-4})(t^{28})^{-4} = \frac{1}{4^4} t^{-112} \] Since \(4 = 2^2\), we have \(4^4 = (2^2)^4 = 2^8\), thus: \[ = \frac{1}{2^8} t^{-112} \] This is not equivalent to \(\frac{1}{256} t^{28}\).
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Expression: \((256 t^{-7})^{-4}\) \[ = (256)^{-4} (t^{-7})^{-4} = \frac{1}{256^4} t^{28} \] Now, since \(256 = 2^8\), we have \(256^4 = (2^8)^4 = 2^{32}\): \[ = \frac{1}{2^{32}} t^{28} \] We know that \(\frac{1}{2^8} = \frac{1}{256}\), so: \[ = \frac{1}{256} t^{28} \] This matches \(\frac{1}{256} t^{28}\).
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Expression: \((4 t^{-7})^{-4}\) \[ = 4^{-4} (t^{-7})^{-4} = \frac{1}{4^4} t^{28} \] As before, since \(4^4 = 2^8\), we have: \[ = \frac{1}{2^8} t^{28} = \frac{1}{256} t^{28} \] But since we already have an expression that simplified to \(\frac{1}{256} t^{28}\), we can consider this as well.
The expressions that lead us to \(\frac{1}{256} t^{28}\) are:
- \((256 t^{-7})^{-4}\)
- \((4 t^{-7})^{-4}\)
However, the first expression we simplified, \((256 t^{-7})^{-4}\), gives the first simplified form directly.
Final Answer: \((256 t^{-7})^{-4}\)