Peters Olson
08/21/2024 · Elementary School
Multiply the complex numbers and simplify the result. (Express your answer in the form \( (a+b r) \). \( ) \) \( (3+47)(5-2 \pi) \)
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Tutor-Verified Answer
Step-by-step Solution
To multiply the complex numbers \( (3 + 47) \) and \( (5 - 2\pi) \), we can treat \( 3 + 47 \) as a complex number where \( 3 \) is the real part and \( 47 \) is the imaginary part. Thus, we can rewrite it as \( 3 + 47i \).
Now, we will multiply \( (3 + 47i) \) and \( (5 - 2\pi) \):
\[
(3 + 47i)(5 - 2\pi)
\]
Using the distributive property (also known as the FOIL method for binomials), we have:
\[
= 3 \cdot 5 + 3 \cdot (-2\pi) + 47i \cdot 5 + 47i \cdot (-2\pi)
\]
Calculating each term:
1. \( 3 \cdot 5 = 15 \)
2. \( 3 \cdot (-2\pi) = -6\pi \)
3. \( 47i \cdot 5 = 235i \)
4. \( 47i \cdot (-2\pi) = -94\pi i \)
Now, we can combine these results:
\[
= 15 - 6\pi + 235i - 94\pi i
\]
Next, we combine the real parts and the imaginary parts:
Real part:
\[
15 - 6\pi
\]
Imaginary part:
\[
(235 - 94\pi)i
\]
Thus, the final result is:
\[
(15 - 6\pi) + (235 - 94\pi)i
\]
Expressing this in the form \( (a + br) \), we have:
\[
(15 - 6\pi) + (235 - 94\pi)i
\]
So, the answer is:
\[
(15 - 6\pi) + (235 - 94\pi)i
\]
Quick Answer
\( (15 - 6\pi) + (235 - 94\pi)i \)
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