If the expression 410 was simplified using the Power Rule of Exponents, what is an equivalent expanded expression? (1 point)

Question

Responses

(412)2
left parenthesis 4 superscript 12 baseline right parenthesis squared

(42)8
left parenthesis 4 squared right parenthesis superscript 8 baseline

(45)2
left parenthesis 4 superscript 5 baseline right parenthesis squared

4⋅4⋅4⋅4⋅4⋅4⋅4

Answer 1

To simplify the expression \( 4^{10} \) using the Power Rule of Exponents, we can express it in a different way. The Power Rule of Exponents states that \( (a^m)^n = a^{m \cdot n} \).

To find an equivalent expanded expression for \( 4^{10} \):

1. Write \( 4^{10} \) in a form that can be expanded.
2. The most straightforward expansion is to note that \( 4^{10} \) means multiplying 4 by itself 10 times.

However, we can also express it using the options provided.

Now let's analyze the options given:

1. \( (4^{12})^2 \): This means \( 4^{12 \cdot 2} = 4^{24} \), which is not equivalent to \( 4^{10} \).
2. \( (4^2)^8 \): This means \( 4^{2 \cdot 8} = 4^{16} \), which is also not equivalent to \( 4^{10} \).
3. \( (4^5)^2 \): This means \( 4^{5 \cdot 2} = 4^{10} \), which is equivalent to \( 4^{10} \).
4. \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \): This would represent \( 4 \) multiplied by itself 7 times, which equals \( 4^7 \), not \( 4^{10} \).

Therefore, the correct answer that represents \( 4^{10} \) is:

(4^5)^2

Answer 2

That's wrong

Answer 3

I apologize for the confusion. Let's carefully analyze the original expression \( 4^{10} \) and the options provided again to determine the correct equivalent expanded expression.

The expression \( 4^{10} \) means we multiply 4 by itself 10 times, which can be written as:

\[ 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \]

Now, assessing the options:

1. \( (4^{12})^2 \): This equals \( 4^{24} \) and is not equivalent to \( 4^{10} \).

2. \( (4^{2})^8 \): This equals \( 4^{16} \) and is not equivalent to \( 4^{10} \).

3. \( (4^5)^2 \): This equals \( 4^{10} \), so this option is indeed equivalent to \( 4^{10} \).

4. \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \): This represents \( 4^7 \) which is not equivalent to \( 4^{10} \).

Based on the assessment, the correct equivalent expression for \( 4^{10} \) in expanded form is indeed:

• \( (4^5)^2 \)

This option satisfies the equivalence to \( 4^{10} \) when simplified using the Power Rule, and I stand by my initial judgment regarding the equivalency to the expression. If there's a different angle or context you'd like me to consider, please let me know.

Answer 4

oh yeah sorry I thought that said something else my bad bot <3

Answer 5

No problem at all! Mistakes happen, and I'm here to help. If you have any more questions or need further clarification on anything else, feel free to ask! 😊❤️

Answer 6

May I ask how you got the answer step by step?

Answer 7

Of course! Let's break down the process step by step to understand how we arrived at the equivalent expression for \( 4^{10} \).

Step 1: Understand the Original Expression

The expression given is \( 4^{10} \), which means that 4 is multiplied by itself 10 times.

Step 2: Explore the Power Rule of Exponents

The Power Rule states that when you raise a power to another power, you multiply the exponents: \[ (a^m)^n = a^{m \cdot n} \] This rule allows us to rewrite exponents in a different form.

Step 3: Analyze the Provided Options

We have the following options to evaluate:

1. \( (4^{12})^2 \):

   • Using the Power Rule: \[ (4^{12})^2 = 4^{12 \cdot 2} = 4^{24} \]
   • This is not equal to \( 4^{10} \).

2. \( (4^2)^8 \):

   • Using the Power Rule: \[ (4^2)^8 = 4^{2 \cdot 8} = 4^{16} \]
   • This is also not equal to \( 4^{10} \).

3. \( (4^5)^2 \):

   • Using the Power Rule: \[ (4^5)^2 = 4^{5 \cdot 2} = 4^{10} \]
   • This is equal to \( 4^{10} \), making it a potential equivalent expression.

4. \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \):

   • This expression represents multiplying 4 by itself 7 times: \[ 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 = 4^7 \]
   • This is also not equal to \( 4^{10} \).

Step 4: Identify the Correct Answer

After evaluating all the options, the one that simplifies correctly to \( 4^{10} \) is:

• \( (4^5)^2 \), because when simplified using the Power Rule, it results in \( 4^{10} \).

Conclusion

Thus, the correct and equivalent expanded expression for \( 4^{10} \) is \( (4^{5})^2 \).

If there's anything more specific you'd like to know or if you have more questions, feel free to ask!