Question
If the expression 410 was simplified using the Power Rule of Exponents, what is an equivalent expanded expression? (1 point)%0D%0AResponses%0D%0A%0D%0A4⋅4⋅4⋅4⋅4⋅4⋅4%0D%0A4 times 4 times 4 times 4 times 4 times 4 times 4%0D%0A%0D%0A(45)2%0D%0Aleft parenthesis 4 superscript 5 baseline right parenthesis squared%0D%0A%0D%0A(42)8%0D%0Aleft parenthesis 4 squared right parenthesis superscript 8 baseline%0D%0A%0D%0A(412)2
Answers
GPT-4o mini
To simplify the expression \( 4^{10} \) using the Power Rule of Exponents, we need to represent it in an expanded form. The Power Rule states that \( a^{m \cdot n} = (a^m)^n \).
Thus, \( 4^{10} \) can be expressed as \( (4^5)^2 \) because \( 5 \cdot 2 = 10 \). With this in mind, the equivalent expanded expression if we break down \( 4^{10} \) is:
- **Expanded form**: \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \) (which represents \( 4 \) multiplied by itself 10 times).
The provided options have the following potential representations:
1. \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \) - This is the expanded expression representing \( 4^{7} \).
2. \( 4 \text{ times } 4 \text{ times } 4 \text{ times } 4 \text{ times } 4 \text{ times } 4 \text{ times } 4 \) - This is also an expanded form but less formal.
3. \( (4^5)^2 \) - Correct representation using power rule which equals \( 4^{10} \).
4. \( (4^2)^8 \) - Also equals \( 4^{16} \).
5. \( (4^{12})^2 \) - Equal to \( 4^{24} \).
So, the answer that corresponds to \( 4^{10} \) simplified is \( (4^5)^2 \).
Therefore, the equivalent expanded expression is:
**Correct Answer**: \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \).
Thus, \( 4^{10} \) can be expressed as \( (4^5)^2 \) because \( 5 \cdot 2 = 10 \). With this in mind, the equivalent expanded expression if we break down \( 4^{10} \) is:
- **Expanded form**: \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \) (which represents \( 4 \) multiplied by itself 10 times).
The provided options have the following potential representations:
1. \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \) - This is the expanded expression representing \( 4^{7} \).
2. \( 4 \text{ times } 4 \text{ times } 4 \text{ times } 4 \text{ times } 4 \text{ times } 4 \text{ times } 4 \) - This is also an expanded form but less formal.
3. \( (4^5)^2 \) - Correct representation using power rule which equals \( 4^{10} \).
4. \( (4^2)^8 \) - Also equals \( 4^{16} \).
5. \( (4^{12})^2 \) - Equal to \( 4^{24} \).
So, the answer that corresponds to \( 4^{10} \) simplified is \( (4^5)^2 \).
Therefore, the equivalent expanded expression is:
**Correct Answer**: \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \).
u sure
GPT-4o mini
I apologize for any confusion in my previous response. Let's clarify the original question regarding \( 4^{10} \).
The expression \( 4^{10} \) means \( 4 \) multiplies itself \( 10 \) times:
- The correct expanded form of \( 4^{10} \) is indeed:
\[
4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4
\]
If we focus on the context of the options provided in your original question, we see that none of the options simply expand to \( 4^{10} \).
Let’s also correct and assess the Power Rule mention. The expressions using the power rule would not apply to deriving an expanded form directly but rather could re-express \( 4^{10} \) itself.
Here is a breakdown again:
1. **\( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \)** - This is the correct expanded form that directly reflects \( 4^{10} \).
2. **\( (4^5)^2 \)** - This is equal to \( 4^{10} \) using the power of a power rule.
3. **\( (4^2)^8 \)** - This simplifies to \( 4^{16} \), which is not \( 4^{10} \).
4. **\( (4^{12})^2 \)** - This results in \( 4^{24} \), also not \( 4^{10} \).
Given these points, the expression \( 4^{10} \) correctly simplifies to either the expanded multiplication stated, or any accurate exponentiation representation like \( (4^5)^2 \) reflecting the Algebraic identity.
For ultimate clarity based on your initial question, the best equivalent **expanded expression** is simply:
**Expanded form**: \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \).
Thank you for your patience, and I appreciate the opportunity to clarify!
The expression \( 4^{10} \) means \( 4 \) multiplies itself \( 10 \) times:
- The correct expanded form of \( 4^{10} \) is indeed:
\[
4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4
\]
If we focus on the context of the options provided in your original question, we see that none of the options simply expand to \( 4^{10} \).
Let’s also correct and assess the Power Rule mention. The expressions using the power rule would not apply to deriving an expanded form directly but rather could re-express \( 4^{10} \) itself.
Here is a breakdown again:
1. **\( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \)** - This is the correct expanded form that directly reflects \( 4^{10} \).
2. **\( (4^5)^2 \)** - This is equal to \( 4^{10} \) using the power of a power rule.
3. **\( (4^2)^8 \)** - This simplifies to \( 4^{16} \), which is not \( 4^{10} \).
4. **\( (4^{12})^2 \)** - This results in \( 4^{24} \), also not \( 4^{10} \).
Given these points, the expression \( 4^{10} \) correctly simplifies to either the expanded multiplication stated, or any accurate exponentiation representation like \( (4^5)^2 \) reflecting the Algebraic identity.
For ultimate clarity based on your initial question, the best equivalent **expanded expression** is simply:
**Expanded form**: \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \).
Thank you for your patience, and I appreciate the opportunity to clarify!