To simplify the expression \( 7^{15} \) using the Power Rule of Exponents and provide an equivalent expanded expression, we first recognize that \( 7^{15} \) can be thought of as \( (7^5)^3 \).
However, let's analyze the given options:
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\( (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \):
- This represents \( 7^{3 \cdot 5} = 7^{15} \), which is a correct expanded form equivalent to \( 7^{15} \).
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\( (7^{10})^5 \):
- Using the Power Rule, this simplifies to \( 7^{10 \times 5} = 7^{50} \), which is not equivalent to \( 7^{15} \).
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\( (7 \cdot 7 \cdot 7 \cdot 7 \cdot 7)^{10} \):
- This represents \( (7^5)^{10} = 7^{5 \times 10} = 7^{50} \), which is also not equivalent to \( 7^{15} \).
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\( (7^{18})^3 \):
- This simplifies to \( 7^{18 \times 3} = 7^{54} \), which is also not equivalent to \( 7^{15} \).
The correct and equivalent expanded expression from the provided options is:
\( (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \).