To evaluate the expression [tex]\((2 - 5i)(p + q)(i)\)[/tex] given that [tex]\(p = 2\)[/tex] and [tex]\(q = 5i\)[/tex], we follow these steps:
1. Substitute the values of [tex]\(p\)[/tex] and [tex]\(q\)[/tex] into the expression [tex]\( (2 - 5i)(p + q) \)[/tex]:
Given:
[tex]\[
p = 2
\][/tex]
[tex]\[
q = 5i
\][/tex]
Substitute [tex]\(p\)[/tex] and [tex]\(q\)[/tex] into the expression:
[tex]\[
(2 - 5i)(2 + 5i)
\][/tex]
2. Calculate [tex]\((2 - 5i)(2 + 5i)\)[/tex]:
We need to use the distributive property (also known as the FOIL method for binomials) to expand this:
[tex]\[
(2 - 5i)(2 + 5i) = 2 \cdot 2 + 2 \cdot 5i - 5i \cdot 2 - 5i \cdot 5i
\][/tex]
Simplify each term:
[tex]\[
= 4 + 10i - 10i - 25i^2
\][/tex]
Recall that [tex]\( i^2 = -1 \)[/tex]:
[tex]\[
= 4 + 10i - 10i + 25
\][/tex]
[tex]\[
= 4 + 25
\][/tex]
[tex]\[
= 29
\][/tex]
3. Multiply the result by [tex]\(i\)[/tex]:
Now we have the intermediate result, [tex]\(29\)[/tex], and we need to multiply this by [tex]\(i\)[/tex]:
[tex]\[
29 \cdot i = 29i
\][/tex]
Thus, the value of [tex]\((2 - 5i)(p + q)(i)\)[/tex] when [tex]\(p = 2\)[/tex] and [tex]\(q = 5i\)[/tex] is [tex]\(\boxed{29i}\)[/tex].
