Answer :
To determine which of the given options is equivalent to the expression \( f(x) = 440 \cdot (2)^{\frac{x}{12}} \), we should evaluate each provided option methodically.
1. Option 1: \( 440 \cdot \times \sqrt{2^{12}} \)
This option implies a calculation involving a square root, but it does not correspond to the given formula.
2. Option 2: \( 440 \cdot 12 \sqrt{2^x} \)
This option suggests multiplying by 12 and taking the square root of \(2^x\). This does not match our formula because:
- The given formula has \( 2^{\frac{x}{12}} \) while this option implies \( \sqrt{2^x} = (2^x)^{\frac{1}{2}} \), which is different.
3. Option 3: \( 440 \cdot \sqrt[12]{2^x} \)
This option suggests taking the 12th root of \(2^x\). Let's simplify this:
[tex]\[ \sqrt[12]{2^x} = (2^x)^{\frac{1}{12}} \][/tex]
Using the properties of exponents, \((2^x)^{\frac{1}{12}} = 2^{\frac{x}{12}}\).
Therefore:
[tex]\[ 440 \cdot \sqrt[12]{2^x} = 440 \cdot 2^{\frac{x}{12}} \][/tex]
This matches the given formula.
4. Option 4: \( 440 \cdot \sqrt[x]{2^{12}} \)
This option implies taking the x-th root of \(2^{12}\). Simplifying this:
[tex]\[ \sqrt[x]{2^{12}} = (2^{12})^{\frac{1}{x}} \][/tex]
This is not equal to \( 2^{\frac{x}{12}} \)...
From the above evaluations, it is clear that:
[tex]\[ \boxed{440 \cdot \sqrt[12]{2^x}} \][/tex]
is the correct and equivalent expression to [tex]\( f(x) = 440 \cdot (2)^{\frac{x}{12}} \)[/tex].
1. Option 1: \( 440 \cdot \times \sqrt{2^{12}} \)
This option implies a calculation involving a square root, but it does not correspond to the given formula.
2. Option 2: \( 440 \cdot 12 \sqrt{2^x} \)
This option suggests multiplying by 12 and taking the square root of \(2^x\). This does not match our formula because:
- The given formula has \( 2^{\frac{x}{12}} \) while this option implies \( \sqrt{2^x} = (2^x)^{\frac{1}{2}} \), which is different.
3. Option 3: \( 440 \cdot \sqrt[12]{2^x} \)
This option suggests taking the 12th root of \(2^x\). Let's simplify this:
[tex]\[ \sqrt[12]{2^x} = (2^x)^{\frac{1}{12}} \][/tex]
Using the properties of exponents, \((2^x)^{\frac{1}{12}} = 2^{\frac{x}{12}}\).
Therefore:
[tex]\[ 440 \cdot \sqrt[12]{2^x} = 440 \cdot 2^{\frac{x}{12}} \][/tex]
This matches the given formula.
4. Option 4: \( 440 \cdot \sqrt[x]{2^{12}} \)
This option implies taking the x-th root of \(2^{12}\). Simplifying this:
[tex]\[ \sqrt[x]{2^{12}} = (2^{12})^{\frac{1}{x}} \][/tex]
This is not equal to \( 2^{\frac{x}{12}} \)...
From the above evaluations, it is clear that:
[tex]\[ \boxed{440 \cdot \sqrt[12]{2^x}} \][/tex]
is the correct and equivalent expression to [tex]\( f(x) = 440 \cdot (2)^{\frac{x}{12}} \)[/tex].
