We are given the formula to determine the frequency [tex]\( f \)[/tex] for the note G above A440, which is:
[tex]\[ f = 440 \cdot 2^{\frac{10}{12}} \][/tex]
We need to determine which of the given options is equivalent to this expression.
1. Option 1: [tex]\( 440 \cdot 10 \sqrt{2^{12}} \)[/tex]
Let's evaluate this option step by step:
[tex]\[
440 \cdot 10 \sqrt{2^{12}}
\][/tex]
Since ~[tex]\( \sqrt{2^{12}} = 2^6 \)[/tex], we can rewrite it as:
[tex]\[
440 \cdot 10 \cdot 2^6 = 440 \cdot 10 \cdot 64 = 281600
\][/tex]
This value does not match [tex]\( f = 440 \cdot 2^{\frac{10}{12}} \)[/tex].
2. Option 2: [tex]\( 440 \cdot 12 \sqrt{2^{10}} \)[/tex]
Evaluating this option:
[tex]\[
440 \cdot 12 \sqrt{2^{10}}
\][/tex]
Since [tex]\( \sqrt{2^{10}} = 2^5 \)[/tex], we rewrite it as:
[tex]\[
440 \cdot 12 \cdot 2^5 = 440 \cdot 12 \cdot 32 = 168960
\][/tex]
This value also does not match [tex]\( f \)[/tex].
3. Option 3: [tex]\( 440 \cdot \sqrt[10]{2^{12}} \)[/tex]
Let's evaluate this:
[tex]\[
440 \cdot \sqrt[10]{2^{12}}
\][/tex]
Which is:
[tex]\[
440 \cdot 2^{\frac{12}{10}} = 440 \cdot 2^{1.2}
\][/tex]
Rewriting [tex]\( 2^{1.2} \)[/tex] into another form, we find:
[tex]\[
2^{1.2} = (2^{12})^{\frac{1}{10}}
\][/tex]
Therefore:
[tex]\[
440 \cdot (2^{1.2}) = 440 \cdot (2^{12})^{\frac{1}{10}} = 1010.8545523973909
\][/tex]
This value does not match [tex]\( f \)[/tex].
4. Option 4: [tex]\( 440 \cdot \sqrt[12]{2^{10}} \)[/tex]
Let's evaluate this last option:
[tex]\[
440 \cdot \sqrt[12]{2^{10}}
\][/tex]
Which simplifies to:
[tex]\[
440 \cdot (2^{10})^{\frac{1}{12}} = 440 \cdot 2^{\frac{10}{12}}
\][/tex]
This is precisely our given formula:
[tex]\[
440 \cdot 2^{\frac{10}{12}} \approx 783.9908719634985
\][/tex]
Thus, the correct option that matches the given formula [tex]\( f = 440 \cdot 2^{\frac{10}{12}} \)[/tex] is:
[tex]\[ \boxed{440 \cdot \sqrt[12]{2^{10}}} \][/tex]
