Question
If the expression 715 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 (710)5 left parenthesis 7 superscript 10 baseline right parenthesis superscript 5 baseline
Answers
GPT-4o mini
To simplify the expression \( 7^{15} \) using the Power Rule of Exponents, we can recognize that \( 15 \) can be broken down into smaller components involving powers of \( 7 \).
The expression \( 7^{15} \) can be represented as:
- \( (7^5)^3 \), since \( 5 \cdot 3 = 15 \).
We can expand this:
- \( (7^5)^3 = (7 \cdot 7 \cdot 7 \cdot 7 \cdot 7)(7 \cdot 7 \cdot 7 \cdot 7 \cdot 7)(7 \cdot 7 \cdot 7 \cdot 7 \cdot 7) \),
which consists of three groups of \( 7^5 \).
Now, looking at the options provided:
1. \( (7 \cdot 7 \cdot 7 \cdot 7 \cdot 7)^{10} \) -> This is \( 7^{50} \), not equivalent.
2. \( (7^{18})^3 \) -> This is \( 7^{54} \), not equivalent.
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 equals \( 7^{15} \) because there are 5 groups of \( 7^3 \), and \( 3 \times 5 = 15 \), hence it is equivalent.
4. \( (7^{10})^5 \) -> This is \( 7^{50} \), not equivalent.
The correct answer is:
**\( (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \)**, since this represents 5 groups of 3 factors of \( 7 \), thereby totaling \( 15 \) factors of \( 7 \).
The expression \( 7^{15} \) can be represented as:
- \( (7^5)^3 \), since \( 5 \cdot 3 = 15 \).
We can expand this:
- \( (7^5)^3 = (7 \cdot 7 \cdot 7 \cdot 7 \cdot 7)(7 \cdot 7 \cdot 7 \cdot 7 \cdot 7)(7 \cdot 7 \cdot 7 \cdot 7 \cdot 7) \),
which consists of three groups of \( 7^5 \).
Now, looking at the options provided:
1. \( (7 \cdot 7 \cdot 7 \cdot 7 \cdot 7)^{10} \) -> This is \( 7^{50} \), not equivalent.
2. \( (7^{18})^3 \) -> This is \( 7^{54} \), not equivalent.
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 equals \( 7^{15} \) because there are 5 groups of \( 7^3 \), and \( 3 \times 5 = 15 \), hence it is equivalent.
4. \( (7^{10})^5 \) -> This is \( 7^{50} \), not equivalent.
The correct answer is:
**\( (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \cdot (7 \cdot 7 \cdot 7) \)**, since this represents 5 groups of 3 factors of \( 7 \), thereby totaling \( 15 \) factors of \( 7 \).