Bot GPT 4-o mini To simplify the expression \( 4^{10} \) using the Power Rule of Exponents, we need to recognize that we can express \( 4^{10} \) in terms of its prime factorization. Specifically, \( 4 \) can be rewritten as \( 4^1 \).
The Power Rule states that \( (a^m)^n = a^{m \cdot n} \). Thus, if we rewrite \( 4 \) in terms of its power, we have \( 4 = 4^1 \).
However, for one of the provided options to be equivalent to \( 4^{10} \), we must check if any transformation matches \( 4^{10} \).
Let's analyze the options:
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\( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \): This translates to \( 4^7 \), which is not \( 4^{10} \).
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\( (4^2)^8 \): Using the Power Rule, this simplifies to \( 4^{2 \cdot 8} = 4^{16} \), which is not \( 4^{10} \).
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\( (4^5)^2 \): This simplifies to \( 4^{5 \cdot 2} = 4^{10} \), which is correct.
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\( (4^{12})^2 \): This simplifies to \( 4^{12 \cdot 2} = 4^{24} \), which is not \( 4^{10} \).
Therefore, the correct choice that is equivalent to \( 4^{10} \) is:
\( (4^5)^2 \).
1 answer