On a piano, the formula to determine the frequency [tex]$f[tex]$[/tex] of any pitch relative to the [tex]$[/tex]A[tex]$[/tex] above middle [tex]$[/tex]C[tex]$[/tex] (A440) is given by the expression [tex]$[/tex]f(x) = 440 \cdot 2^{\frac{x}{12}}[tex]$[/tex], where [tex]$[/tex]x[tex]$[/tex] represents the number of half-steps above or below [tex]$[/tex]A440[tex]$[/tex]. Which of the following is equivalent to [tex]$[/tex]f(x)$[/tex]?

A. [tex]440 \cdot \sqrt{2^{12}}[/tex]

B. [tex]440 \cdot 12 \sqrt{2^x}[/tex]

C. [tex]440 \cdot \sqrt[12]{2^x}[/tex]

D. [tex]440 \cdot \sqrt[x]{2^{12}}[/tex]



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].