Question
Describe the relationship between the number 2 x
and 4 x
.
Responses
A 4 x
is twice 2 x
4 x 10 6 is twice 2 x 10 4
B 4 x
is 20 times more than 2 x
4 x 10 6 is 20 times more than 2 x 10 4
C 4 x
is 200 times more than 2 x
4 x 10 6 is 200 times more than 2 x 10 4
D 4 x
is 2000 times more than 2 x
4 x 10 6 is 2000 times more than 2 x 10 4
E 4 x
is 100 times more than 2 x
4 x 10 6 is 100 times more than 2 x 10 4
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and 4 x
.
Responses
A 4 x
is twice 2 x
4 x 10 6 is twice 2 x 10 4
B 4 x
is 20 times more than 2 x
4 x 10 6 is 20 times more than 2 x 10 4
C 4 x
is 200 times more than 2 x
4 x 10 6 is 200 times more than 2 x 10 4
D 4 x
is 2000 times more than 2 x
4 x 10 6 is 2000 times more than 2 x 10 4
E 4 x
is 100 times more than 2 x
4 x 10 6 is 100 times more than 2 x 10 4
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Answers
GPT-4o mini
To analyze the relationship between \( 2^x \) and \( 4^x \), we can rewrite \( 4^x \) as \( (2^2)^x = 2^{2x} \). Thus, we see that:
\[ 4^x = 2^{2x} \]
This indicates that \( 4^x \) is equivalent to \( (2^x)^2 \).
Now, let's express \( 4^x \) in relation to \( 2^x \):
\[ 4^x = (2^x)^2 \]
This means that \( 4^x \) is the square of \( 2^x \).
Now, let's explore the numerical terms given in the problem:
1. \( 2 \times 10^4 \) can be simplified to \( 20000 \).
2. \( 4 \times 10^6 \) can be simplified to \( 4000000 \).
To find the relationship between \( 4 \times 10^6 \) and \( 2 \times 10^4 \):
\[ 4 \times 10^6 = 4 \times 1000000 = 4000000 \]
and
\[ 2 \times 10^4 = 2 \times 10000 = 20000 \]
Now, we can calculate how many times \( 4 \times 10^6 \) is compared to \( 2 \times 10^4 \):
\[ \frac{4000000}{20000} = 200 \]
Thus, \( 4 \times 10^6 \) is 200 times greater than \( 2 \times 10^4 \).
From the options provided:
- A states that \( 4^x \) is twice \( 2^x \), which is false.
- B states that \( 4^x \) is 20 times more than \( 2^x \), which is false.
- C states that \( 4^x \) is 200 times more than \( 2^x \), which we have confirmed as true.
- D states that \( 4^x \) is 2000 times more than \( 2^x \), which is false.
- E states that \( 4^x \) is 100 times more than \( 2^x \), which is false.
Therefore, the correct answer is:
**C: \( 4^x \) is 200 times more than \( 2^x \) (and \( 4 \times 10^6 \) is 200 times more than \( 2 \times 10^4 \)).**
\[ 4^x = 2^{2x} \]
This indicates that \( 4^x \) is equivalent to \( (2^x)^2 \).
Now, let's express \( 4^x \) in relation to \( 2^x \):
\[ 4^x = (2^x)^2 \]
This means that \( 4^x \) is the square of \( 2^x \).
Now, let's explore the numerical terms given in the problem:
1. \( 2 \times 10^4 \) can be simplified to \( 20000 \).
2. \( 4 \times 10^6 \) can be simplified to \( 4000000 \).
To find the relationship between \( 4 \times 10^6 \) and \( 2 \times 10^4 \):
\[ 4 \times 10^6 = 4 \times 1000000 = 4000000 \]
and
\[ 2 \times 10^4 = 2 \times 10000 = 20000 \]
Now, we can calculate how many times \( 4 \times 10^6 \) is compared to \( 2 \times 10^4 \):
\[ \frac{4000000}{20000} = 200 \]
Thus, \( 4 \times 10^6 \) is 200 times greater than \( 2 \times 10^4 \).
From the options provided:
- A states that \( 4^x \) is twice \( 2^x \), which is false.
- B states that \( 4^x \) is 20 times more than \( 2^x \), which is false.
- C states that \( 4^x \) is 200 times more than \( 2^x \), which we have confirmed as true.
- D states that \( 4^x \) is 2000 times more than \( 2^x \), which is false.
- E states that \( 4^x \) is 100 times more than \( 2^x \), which is false.
Therefore, the correct answer is:
**C: \( 4^x \) is 200 times more than \( 2^x \) (and \( 4 \times 10^6 \) is 200 times more than \( 2 \times 10^4 \)).**