Which two expressions are equivalent to \( 4^{7} \times 4^{-5} \) ? A \( \frac{1}{4^{-2}} \) B \( \frac{1}{4^{2}} \) C \( 4^{-35} \) D \( 4^{-2} \) E \( 4^{2} \) F \( 4^{12} \)
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To simplify \( 4^{7} \times 4^{-5} \), we can use the property of exponents that states \( a^{m} \times a^{n} = a^{m+n} \). So, we have: \[ 4^{7} \times 4^{-5} = 4^{7 + (-5)} = 4^{2} \] Now we are looking for expressions equivalent to \( 4^{2} \). The equivalent expression from the options provided is: E \( 4^{2} \) As for the next equivalent, when looking at the negative exponent property, we find that \( 4^{2} \) can also be expressed as \( \frac{1}{4^{-2}} \). Thus, the other equivalent expression is: A \( \frac{1}{4^{-2}} \) So the two expressions that are equivalent to \( 4^{7} \times 4^{-5} \) are A and E.