To show that $\overline{zw}$ and $\overline{z} \cdot \overline{w}$ are conjugates, we start by expressing $z$ and $w$ in terms of their real and imaginary parts: $z = a + bi$ $w = c + di$ where $a, b, c,$ and $d$ are real numbers, and $i$ is the imaginary unit with $i^2 = -1$.

First, compute $zw$: $zw = (a + bi)(c + di)$

Using the distributive property (FOIL method): $zw = ac + adi + bci + bdi^2$

Since $i^2 = -1$, this simplifies to: $zw = ac + adi + bci + bd(-1)$ $zw = ac + adi + bci - bd$ $zw = (ac - bd) + (ad + bc)i$

Now, compute the conjugate of $zw$, denoted as $\overline{zw}$: $\overline{zw} = \overline{(ac - bd) + (ad + bc)i}$ The conjugate of a complex number $x + yi$ is $x - yi$: $\overline{zw} = (ac - bd) - (ad + bc)i$

Next, compute the conjugates of $z$ and $w$: $\overline{z} = \overline{a + bi} = a - bi$ $\overline{w} = \overline{c + di} = c - di$

Now, compute the product $\overline{z} \cdot \overline{w}$: $\overline{z} \cdot \overline{w} = (a - bi)(c - di)$

Using the distributive property again: $\overline{z} \cdot \overline{w} = ac - adi - bci + bdi^2$

Since $i^2 = -1$, this simplifies to: $\overline{z} \cdot \overline{w} = ac - adi - bci + bd(-1)$ $\overline{z} \cdot \overline{w} = ac - adi - bci - bd$ $\overline{z} \cdot \overline{w} = (ac - bd) - (ad + bc)i$

We see that: $\overline{zw} = (ac - bd) - (ad + bc)i$ $\overline{z} \cdot \overline{w} = (ac - bd) - (ad + bc)i$

Thus: $\overline{zw} = \overline{z} \cdot \overline{w}$

Therefore, $\overline{zw}$ and $\overline{z} \cdot \overline{w}$ are indeed conjugates.

Do you want details or have any questions?

Here are some related questions to expand on this topic:

1. What are the properties of complex number conjugates?
2. How does the modulus of a complex number relate to its conjugate?
3. Can you show that the product of a complex number and its conjugate is always a non-negative real number?
4. How do the conjugates of sums and differences of complex numbers relate to the sums and differences of their conjugates?
5. What is the geometric interpretation of complex conjugation on the complex plane?

Tip: Always remember that the conjugate of a product of two complex numbers is the product of their conjugates.