Let's evaluate the given expression [tex]\((2-5i)(p+q)(i)\)[/tex] with the values [tex]\(p = 2\)[/tex] and [tex]\(q = 5i\)[/tex].
1. Substitute the values:
Substitute [tex]\(p\)[/tex] and [tex]\(q\)[/tex] into the expression:
[tex]\[
(2 - 5i)(2 + 5i)(i)
\][/tex]
2. Combine the complex numbers:
First, calculate [tex]\((2 + 5i)\)[/tex]:
[tex]\[
(2 - 5i)(2 + 5i)
\][/tex]
3. Multiply the complex numbers:
Use the distributive property (FOIL) to simplify:
[tex]\[
(2 - 5i)(2 + 5i) = 2 \cdot 2 + 2 \cdot 5i - 5i \cdot 2 - 5i \cdot 5i
\][/tex]
4. Perform the multiplications:
[tex]\[
= 4 + 10i - 10i - 25i^2
\][/tex]
Recall that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
-25i^2 = -25(-1) = 25
\][/tex]
5. Combine like terms:
[tex]\[
4 + 10i - 10i + 25 = 4 + 25 = 29
\][/tex]
6. Multiply the result with [tex]\(i\)[/tex]:
Now we multiply our result by [tex]\(i\)[/tex]:
[tex]\[
(29)(i) = 29i
\][/tex]
7. Conclusion:
The value of the expression [tex]\((2-5i)(p+q)(i)\)[/tex] when [tex]\(p=2\)[/tex] and [tex]\(q=5i\)[/tex] is [tex]\(29i\)[/tex]. The correct answer is:
[tex]\[
\boxed{29i}
\][/tex]
