To evaluate the expression [tex]\((2 - 5i)(p + q)(i)\)[/tex] given [tex]\(p = 2\)[/tex] and [tex]\(q = 5i\)[/tex], let's proceed with a detailed, step-by-step solution.
1. Substitute the given values of [tex]\(p\)[/tex] and [tex]\(q\)[/tex]:
- [tex]\(p = 2\)[/tex]
- [tex]\(q = 5i\)[/tex]
Substituting these into [tex]\((p + q)\)[/tex], we get:
[tex]\[
p + q = 2 + 5i
\][/tex]
2. First Expression:
- The first expression is already given as [tex]\((2 - 5i)\)[/tex].
3. Combine the expressions:
- Now we have [tex]\((2 - 5i)\)[/tex] and [tex]\((2 + 5i)\)[/tex].
4. Multiply the complex numbers [tex]\((2 - 5i)\)[/tex] and [tex]\((2 + 5i)\)[/tex]:
- When multiplying complex conjugates, the result is a real number:
[tex]\[
(2 - 5i)(2 + 5i) = 2^2 - (5i)^2 = 4 - 25(-1) = 4 + 25 = 29
\][/tex]
- So, [tex]\((2 - 5i)(2 + 5i) = 29\)[/tex].
5. Multiply the result by [tex]\(i\)[/tex]:
- Now, we need to multiply the above result by [tex]\(i\)[/tex]:
[tex]\[
29 \times i = 29i
\][/tex]
Thus, the value of [tex]\((2 - 5i)(2 + 5i)(i)\)[/tex] is [tex]\(\boxed{29i}\)[/tex].
