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Sagot :
To solve the problem, we need to determine the correct expression equivalent to the given formula for the frequency, [tex]\( f \)[/tex], of the G above A440:
[tex]\[ f = 440 \cdot 2^{\frac{10}{12}} \][/tex]
We are given four multiple choice options and need to evaluate if any of these match the original formula. Here is each option analyzed individually:
### Option 1: [tex]\( 440 \cdot 10 \sqrt{2^{12}} \)[/tex]
First, let's simplify this expression:
[tex]\[ 10 \sqrt{2^{12}} = 10 \cdot \sqrt{2^{12}} = 10 \cdot 2^6 = 10 \cdot 64 = 640 \][/tex]
Thus, this expression becomes:
[tex]\[ 440 \cdot 640 = 281600 \][/tex]
This is not equivalent to the formula [tex]\( f = 440 \cdot 2^{\frac{10}{12}} \)[/tex].
### Option 2: [tex]\( 440 \cdot 12 \sqrt{2^{10}} \)[/tex]
Next, let's simplify this expression:
[tex]\[ 12 \sqrt{2^{10}} = 12 \cdot \sqrt{2^{10}} = 12 \cdot 2^5 = 12 \cdot 32 = 384 \][/tex]
Thus, this expression becomes:
[tex]\[ 440 \cdot 384 = 168960 \][/tex]
This is also not equivalent to the formula [tex]\( f = 440 \cdot 2^{\frac{10}{12}} \)[/tex].
### Option 3: [tex]\( 440 \cdot \sqrt[10]{2^{12}} \)[/tex]
Now, let's simplify this expression:
[tex]\[ \sqrt[10]{2^{12}} = (2^{12})^{\frac{1}{10}} = 2^{\frac{12}{10}} = 2^{1.2} \][/tex]
Thus, this expression becomes:
[tex]\[ 440 \cdot 2^{1.2} \][/tex]
Although the exponent appears similar to [tex]\( 2^{\frac{10}{12}} \)[/tex], the numerical evaluation:
[tex]\[ 2^{\frac{12}{10}} \approx 2^{1.2} \][/tex]
Is not the same as [tex]\( 2^{\frac{10}{12}} \)[/tex], so this option is not equivalent.
### Option 4: [tex]\( 440 \cdot \sqrt[12]{2^{10}} \)[/tex]
Finally, let's simplify this expression:
[tex]\[ \sqrt[12]{2^{10}} = (2^{10})^{\frac{1}{12}} = 2^{\frac{10}{12}} \][/tex]
Thus, this expression becomes:
[tex]\[ 440 \cdot 2^{\frac{10}{12}} \][/tex]
This matches the original formula exactly. Therefore, option 4:
[tex]\[ 440 \cdot \sqrt[12]{2^{10}} \][/tex]
is equivalent to the given formula [tex]\( f = 440 \cdot 2^{\frac{10}{12}} \)[/tex].
### Conclusion:
The correct answer is:
[tex]\[ 440 \cdot \sqrt[12]{2^{10}} \][/tex]
[tex]\[ f = 440 \cdot 2^{\frac{10}{12}} \][/tex]
We are given four multiple choice options and need to evaluate if any of these match the original formula. Here is each option analyzed individually:
### Option 1: [tex]\( 440 \cdot 10 \sqrt{2^{12}} \)[/tex]
First, let's simplify this expression:
[tex]\[ 10 \sqrt{2^{12}} = 10 \cdot \sqrt{2^{12}} = 10 \cdot 2^6 = 10 \cdot 64 = 640 \][/tex]
Thus, this expression becomes:
[tex]\[ 440 \cdot 640 = 281600 \][/tex]
This is not equivalent to the formula [tex]\( f = 440 \cdot 2^{\frac{10}{12}} \)[/tex].
### Option 2: [tex]\( 440 \cdot 12 \sqrt{2^{10}} \)[/tex]
Next, let's simplify this expression:
[tex]\[ 12 \sqrt{2^{10}} = 12 \cdot \sqrt{2^{10}} = 12 \cdot 2^5 = 12 \cdot 32 = 384 \][/tex]
Thus, this expression becomes:
[tex]\[ 440 \cdot 384 = 168960 \][/tex]
This is also not equivalent to the formula [tex]\( f = 440 \cdot 2^{\frac{10}{12}} \)[/tex].
### Option 3: [tex]\( 440 \cdot \sqrt[10]{2^{12}} \)[/tex]
Now, let's simplify this expression:
[tex]\[ \sqrt[10]{2^{12}} = (2^{12})^{\frac{1}{10}} = 2^{\frac{12}{10}} = 2^{1.2} \][/tex]
Thus, this expression becomes:
[tex]\[ 440 \cdot 2^{1.2} \][/tex]
Although the exponent appears similar to [tex]\( 2^{\frac{10}{12}} \)[/tex], the numerical evaluation:
[tex]\[ 2^{\frac{12}{10}} \approx 2^{1.2} \][/tex]
Is not the same as [tex]\( 2^{\frac{10}{12}} \)[/tex], so this option is not equivalent.
### Option 4: [tex]\( 440 \cdot \sqrt[12]{2^{10}} \)[/tex]
Finally, let's simplify this expression:
[tex]\[ \sqrt[12]{2^{10}} = (2^{10})^{\frac{1}{12}} = 2^{\frac{10}{12}} \][/tex]
Thus, this expression becomes:
[tex]\[ 440 \cdot 2^{\frac{10}{12}} \][/tex]
This matches the original formula exactly. Therefore, option 4:
[tex]\[ 440 \cdot \sqrt[12]{2^{10}} \][/tex]
is equivalent to the given formula [tex]\( f = 440 \cdot 2^{\frac{10}{12}} \)[/tex].
### Conclusion:
The correct answer is:
[tex]\[ 440 \cdot \sqrt[12]{2^{10}} \][/tex]
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