To determine the equivalent expanded expression, we first need to apply the Power Rule of Exponents correctly:
The Power Rule states that \( (a^m)^n = a^{m \cdot n} \).
Let's analyze the expressions provided:
- \( (4^2)^8 \) - According to the Power Rule, this simplifies to \( 4^{2 \cdot 8} = 4^{16} \).
- \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \) - This is \( 4^7 \), since it is multiplying 4 a total of 7 times.
- \( (4^5)^2 \) - This simplifies to \( 4^{5 \cdot 2} = 4^{10} \).
- \( (4^{12})^2 \) - This simplifies to \( 4^{12 \cdot 2} = 4^{24} \).
From the original information, we can discern the expanded forms:
- For \( 4^{16} \): \( 4 \) appears 16 times: \( 4 \times 4 \times 4 \times \ldots \) (16 times)
- For \( 4^7 \): \( 4 \) appears 7 times: \( 4 \times 4 \times \ldots \) (7 times)
- For \( 4^{10} \): \( 4 \) appears 10 times: \( 4 \times 4 \times \ldots \) (10 times)
- For \( 4^{24} \): \( 4 \) appears 24 times: \( 4 \times 4 \times \ldots \) (24 times)
Please clarify which specific expression you are referring to, or if you would like to see the expanded version of a specific exponentiation.