Simplify the following expression.

(4−7i)(4+7i)


To simplify the expression \((4 - 7i)(4 + 7i)\), we can use the difference of squares formula:

\[
(a - b)(a + b) = a^2 - b^2
\]

Here, \(a = 4\) and \(b = 7i\). Applying the formula, we get:

\[
(4 - 7i)(4 + 7i) = 4^2 - (7i)^2
\]

First, compute \(4^2\):

\[
4^2 = 16
\]

Next, compute \((7i)^2\):

\[
(7i)^2 = 7^2 \cdot i^2 = 49 \cdot (-1) = -49
\]

Now, substitute these values back into the expression:

\[
(4 - 7i)(4 + 7i) = 16 - (-49)
\]

Simplify the expression:

\[
16 - (-49) = 16 + 49 = 65
\]

Therefore, the simplified expression is:

\[
(4 - 7i)(4 + 7i) = 65
\]

Would you like any further details or have any questions?

### Further Questions:
1. How does the difference of squares formula apply to complex numbers?
2. Can you simplify another expression using the difference of squares?
3. What is the significance of the imaginary unit \(i\)?
4. How do you multiply two complex numbers in general?
5. Can you provide an example where the difference of squares formula doesn't apply?
6. How can you check the correctness of the simplified result?
7. What is the geometric interpretation of multiplying complex conjugates?
8. What happens when you multiply a complex number by its conjugate?

### Tip:
When multiplying complex conjugates, remember that the imaginary parts cancel out, resulting in a real number.

Learn how to simplify the expression (4-7i)(4+7i) step-by-step using the difference of squares formula. Understand the application of complex numbers and the formula used. Suitable for advanced high school students.

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