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Simplify:
[tex]\[
(7-5i)(7+5i)
\][/tex]

A. 14
B. 24
C. 74
D. [tex]\(49-25i\)[/tex]
E. [tex]\(49-70i-25i^2\)[/tex]

Sagot :

GinnyAnsweridn
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
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