If the expression 715 was simplified using the Power Rule of Exponents, which of the following would be an expanded equivalent expression?(1 point)

(718)^3

(7⋅7⋅7⋅7⋅7)^10

(7^10)^5

(7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7)

1 answer

To simplify the expression \( 7^{15} \) using the Power Rule of Exponents, we should express it in a way that reveals its expanded form.

The Power Rule states that \( (a^m)^n = a^{m \cdot n} \).

Let's analyze the options:

  1. \( (7^{18})^3 \): This simplifies to \( 7^{18 \cdot 3} = 7^{54} \), which is not equivalent to \( 7^{15} \).

  2. \( (7 \cdot 7 \cdot 7 \cdot 7 \cdot 7)^{10} \): This equals \( (7^5)^{10} = 7^{5 \cdot 10} = 7^{50} \), which is not equivalent to \( 7^{15} \).

  3. \( (7^{10})^5 \): This simplifies to \( 7^{10 \cdot 5} = 7^{50} \), which is not equivalent to \( 7^{15} \).

  4. \( (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \): This represents \( 5 \) groups of \( (7 \cdot 7 \cdot 7) = 7^3 \). Hence, this expands to \( (7^3)^5 = 7^{3 \cdot 5} = 7^{15} \), which is equivalent to \( 7^{15} \).

Therefore, the correct answer is:

(7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7).