To simplify the expression \( 410 \) using the Power Rule of Exponents, you can express it in an expanded form by interpreting \( 410 \) as \( 4^{10} \).
Using the Power Rule of Exponents, \( a^m \) can be expressed as \( (a^n)^k \) where \( m = n \cdot k \).
The equivalent expanded expression for \( 4^{10} \) is:
- Expanded form: \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \)
However, looking at your multiple choice responses:
- \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \) corresponds to \( 4^7\) which is not correct for \( 4^{10} \).
- \( (45)^2 \) represents \( 45^2 \) not \( 4^{10} \).
- \( (42)^8 \) represents \( 42^8 \) not \( 4^{10} \).
- \( (4^{12})^2 \) simplifies to \( 4^{24} \) not \( 4^{10} \).
It seems that none of the options provided are entirely correct for \( 4^{10} \).
The expanded expression equivalent to \( 4^{10} \) would be explicitly written as: \[ 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \]
If you are looking for a correct option based on your original choices, it appears that there might be a misunderstanding or misrepresentation of the answers given, since none match \( 4^{10} \).
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