To simplify the expression \( 4^{10} \) using the Power Rule of Exponents, we can rewrite it as follows:
- We know that \( (a^m)^n = a^{m \cdot n} \). Therefore, we can express \( 4^{10} \) in forms like \( (4^2)^5 \) or \( (4^5)^2 \) since \( 2 \cdot 5 = 10 \) and \( 5 \cdot 2 = 10 \).
Considering the options provided:
- \( (4^2)^8 \) simplifies to \( 4^{2 \cdot 8} = 4^{16} \) (not equivalent).
- \( (4^5)^2 \) simplifies to \( 4^{5 \cdot 2} = 4^{10} \) (this is equivalent).
- \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \) represents \( 4^7 \) (not equivalent).
- \( (4^{12})^2 \) simplifies to \( 4^{12 \cdot 2} = 4^{24} \) (not equivalent).
The equivalent expanded expression of \( 4^{10} \) among the options given is:
(4^5)^2.