To simplify the expression [tex]\((7-5i)(7+5i)\)[/tex], let’s follow the steps:
1. Recognize the form: The expression [tex]\((7-5i)(7+5i)\)[/tex] is in the form of [tex]\((a - bi)(a + bi)\)[/tex], which is a difference of squares. For complex numbers, this can be simplified using the formula:
[tex]\[
(a - bi)(a + bi) = a^2 - (bi)^2
\][/tex]
2. Identify the values: Here [tex]\(a = 7\)[/tex] and [tex]\(b = 5\)[/tex].
3. Apply the values to the formula:
[tex]\[
(7 - 5i)(7 + 5i) = 7^2 - (5i)^2
\][/tex]
4. Square the terms:
- [tex]\(7^2 = 49\)[/tex]
- [tex]\((5i)^2 = 25i^2\)[/tex]
5. Simplify [tex]\(i^2\)[/tex]: Recall that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
25i^2 = 25(-1) = -25
\][/tex]
6. Combine the terms:
[tex]\[
(7 - 5i)(7 + 5i) = 49 - (-25) = 49 + 25 = 74
\][/tex]
So, the simplified form of [tex]\((7-5i)(7+5i)\)[/tex] is [tex]\(74\)[/tex].
Thus, the correct answer is:
C. 74
