For what real number $k$ is $(8 + ki)(7 - 3i)$ equal to a real number?

To determine the real number $ k $ for which the product $(8 + ki)(7 - 3i)$ is a real number, we start by expanding the product using the distributive property:

\[
(8 + ki)(7 - 3i) = 8 \cdot 7 + 8 \cdot (-3i) + ki \cdot 7 + ki \cdot (-3i)
\]

\[
= 56 - 24i + 7ki - 3ki^2
\]

Since $i^2 = -1$, the term $-3ki^2$ becomes $3k$. Substituting this in, we get:

\[
= 56 - 24i + 7ki + 3k
\]

Combining the real and imaginary parts, we have:

\[
= (56 + 3k) + (-24 + 7k)i
\]

For this expression to be a real number, the imaginary part must be zero. Thus, we set the coefficient of $i$ to zero:

\[
-24 + 7k = 0
\]

Solving for $k$:

\[
7k = 24
\]
\[
k = \frac{24}{7}
\]

Thus, the real number $k$ for which $(8 + ki)(7 - 3i)$ is a real number is:

\[
\boxed{\frac{24}{7}}
\]

Would you like more details or have any questions about this solution? Here are some related questions for further exploration:

1. What is the product $(8 + ki)(7 - 3i)$ when $k = \frac{24}{7}$?
2. How would you solve $(8 + ki)(7 - 3i) = 56 + 3k$?
3. What are the general conditions for the product of two complex numbers to be a real number?
4. How does the real part of the product of two complex numbers change with different $k$ values?
5. What is the geometric interpretation of multiplying complex numbers in the complex plane?
6. How does one determine if a given complex number is purely imaginary?
7. What role does the imaginary unit $i$ play in complex number multiplication?
8. How would the result change if $k$ were a complex number instead of a real number?

**Tip:** When multiplying complex numbers, keeping track of the real and imaginary parts separately helps in simplifying the expressions and solving equations efficiently.

Explore how to determine the real number k for which the product (8 + ki)(7 - 3i) results in a real number. Understand the distributive property of complex numbers and conditions for complex numbers to be real. Suitable for high school students learning complex numbers.

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