We simplify the given expression:

$(4 - 7i)(4 + 7i)$

Step 1: Apply the Difference of Squares Formula

We recognize that this is in the form of a difference of squares:

$(a - b)(a + b) = a^2 - b^2$

where $a = 4$ and $b = 7i$. Applying the formula:

$4^2 - (7i)^2$

Step 2: Compute Each Term

$4^2 = 16$ $(7i)^2 = 49i^2$

Since $i^2 = -1$, we substitute:

$49i^2 = 49(-1) = -49$

Step 3: Final Calculation

$16 - (-49) = 16 + 49 = 65$

Final Answer:

$\mathbf{65}$

Would you like a deeper explanation or another example?

Here are five related questions to expand your understanding:

1. Simplify $(3 - 4i)(3 + 4i)$.
2. Compute the modulus of the complex number $4 - 7i$.
3. Expand and simplify $(2 + i)(2 - i)$.
4. What is the general formula for multiplying conjugates of complex numbers?
5. How do you represent complex numbers in polar form?

Tip: The product of a complex number and its conjugate always results in a real number! 🚀