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

(710)5
left parenthesis 7 superscript 10 baseline right parenthesis superscript 5 baseline

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

(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

(718)3

Answer 1

To simplify the expression \( 7^{15} \) using the Power Rule of Exponents, we know that \( a^{m \cdot n} = (a^m)^n \).

To find an expanded equivalent expression for \( 7^{15} \), we can express it as a product of smaller exponential components. The most straightforward way is to express it as \( (7^{5})^3 \), since \( 15 = 5 \times 3 \).

Now, checking the given options to find an equivalent expression:

1. Option 1: \( (7^{10})^{5} \)

   • This expands to \( 7^{10 \cdot 5} = 7^{50} \). Not equivalent.

2. Option 2: \( (7 \cdot 7 \cdot 7 \cdot 7 \cdot 7)^{10} \)

   • This simplifies to \( (7^5)^{10} = 7^{5 \cdot 10} = 7^{50} \). Not equivalent.

3. Option 3: \( (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \)

   • This is equivalent to \( 7^3 \cdot 7^3 \cdot 7^3 \cdot 7^3 \cdot 7^3 = 7^{3 \cdot 5} = 7^{15} \). This is equivalent!

4. Option 4: \( (7^{18})^{3} \)

   • This expands to \( 7^{18 \cdot 3} = 7^{54} \). Not equivalent.

The correct answer is Option 3, which is the only expanded form equivalent to \( 7^{15} \).