To simplify the expression \( \frac{4^{28}}{4} \), we can use the property of exponents that states \( \frac{a^m}{a^n} = a^{m-n} \).
Applying this property, we get:
\[ \frac{4^{28}}{4} = \frac{4^{28}}{4^1} = 4^{28-1} = 4^{27} \]
Now, let's evaluate the given options to see which one is equivalent to \( 4^{27} \):
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Option 1: \( 4^{29} \) - This is not equivalent because \( 4^{29} \) is greater than \( 4^{27} \).
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Option 2: \( \frac{8^{27}}{2^{27}} \) - We can rewrite \( 8 \) in terms of \( 2 \): \[ 8^{27} = (2^3)^{27} = 2^{81} \] So: \[ \frac{8^{27}}{2^{27}} = \frac{2^{81}}{2^{27}} = 2^{81-27} = 2^{54} \] Since \( 4 = 2^2 \), we can express \( 4^{27} \) as \( (2^2)^{27} = 2^{54} \). Therefore, this option is equivalent to \( 4^{27} \).
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Option 3: \( 4^7 \times 4^4 \) - We can use the property of exponents that states \( a^m \times a^n = a^{m+n} \): \[ 4^7 \times 4^4 = 4^{7+4} = 4^{11} \] Thus, this option is not equivalent.
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Option 4: \( 4^{28} \) - This is not equivalent because \( 4^{28} \) is greater than \( 4^{27} \).
From the evaluations above, we find that Option 2: \( \frac{8^{27}}{2^{27}} \) is the only expression that is equivalent to \( 4^{27} \).