If the expression 7^15

Question

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

(7⋅7⋅7⋅7⋅7)^10
left parenthesis 7 times 7 times 7 times 7 times 7 right parenthesis superscript 10 baseline

(718)3
left parenthesis 7 superscript 18 baseline right parenthesis cubed

(7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7)
left parenthesis 7 times 7 times 7 right parenthesis times left parenthesis 7 times 7 times 7 right parenthesis times left parenthesis 7 times 7 times 7 right parenthesis times left parenthesis 7 times 7 times 7 right parenthesis times left parenthesis 7 times 7 times 7 right parenthesis

(7^10)^5

Answer 1

To simplify \( 7^{15} \) using the Power Rule of Exponents, we want to express it in an expanded form that matches the properties of exponents.

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

Let’s analyze the provided options:

1. (7⋅7⋅7⋅7⋅7)^10: This means \( 5 \) multiplied by itself \( 10 \) times, which is \( 7^{5 \cdot 10} = 7^{50} \). This is not equal to \( 7^{15} \).

2. (7^18)^3: This means \( 18 \) multiplied by \( 3 \), which is \( 7^{18 \cdot 3} = 7^{54} \). This is not equal to \( 7^{15} \).

3. (7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7)⋅(7⋅7⋅7): This represents \( 7^3 \) multiplied by itself \( 5 \) times. Therefore, this is \( (7^3)^5 = 7^{3 \cdot 5} = 7^{15} \). This is equal to \( 7^{15} \).

4. (7^10)^5: This represents \( 10 \) multiplied by \( 5 \), which is \( 7^{10 \cdot 5} = 7^{50} \). This is not equal to \( 7^{15} \).

The only correct expanded equivalent expression for \( 7^{15} \) is:

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