*Question E7E08:
What are the principal frequencies that appear at the output of a mixer?*  

*Correct Answer is  C: The two input frequencies along with their sum and difference frequencies*  

### AI Response: (may contain errors)  

In a mixer, the principal frequencies that appear at the output are the result of combining two input frequencies. This process is known as frequency mixing or heterodyning. The correct answer is:

**C: The two input frequencies along with their sum and difference frequencies**

### Explanation:

1. **Correct Answer:**
   - When two signals with frequencies $f_1$ and $f_2$ are mixed, the output contains:
     - The original input frequencies: $f_1$ and $f_2$
     - The sum frequency: $f_1 + f_2$
     - The difference frequency: $|f_1 - f_2|$
   - These are the principal frequencies that appear at the output of a mixer.

2. **Why Other Choices Are Incorrect:**
   - **A: Two and four times the input frequency**
     - This choice suggests harmonics of the input frequencies, which are not the primary result of mixing. Mixers primarily produce sum and difference frequencies, not multiples.
   - **B: The square root of the product of input frequencies**
     - This choice is mathematically incorrect in the context of mixing. Mixers do not produce frequencies based on the square root of the product of input frequencies.
   - **D: 1.414 and 0.707 times the input frequency**
     - These values are related to the square root of 2 and its reciprocal, which are not relevant to the mixing process. Mixers do not inherently produce these specific multiples of the input frequencies.

### Useful Formula:

- The output frequencies of a mixer can be expressed as:
  $$ f_{\text{output}} = \{ f_1, f_2, f_1 + f_2, |f_1 - f_2| \} $$

This formula highlights the key frequencies generated by the mixing process, which are the original frequencies and their sum and difference.