Find the imaginary and real parts of the complex number \((1+2i)^{-2}\).
Answer (2)
The real part of the complex number ( 1 + 2 i ) − 2 is − 25 3 and the imaginary part is − 25 4 .
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To find the real and imaginary parts of the complex number ( 1 + 2 i ) − 2 , we first need to simplify the expression.
Let's start by evaluating ( 1 + 2 i ) − 2 . We know that ( 1 + 2 i ) − 2 = ( 1 + 2 i 1 ) 2 .
To simplify 1 + 2 i 1 , we multiply the numerator and denominator by the conjugate of the denominator, 1 − 2 i . This helps in getting rid of the imaginary unit i in the denominator:
1 + 2 i 1 × 1 − 2 i 1 − 2 i = ( 1 + 2 i ) ( 1 − 2 i ) 1 − 2 i
Calculating the denominator: ( 1 + 2 i ) ( 1 − 2 i ) = 1 2 − ( 2 i ) 2 = 1 − 4 i 2
Since i 2 = − 1 , this becomes: 1 − 4 ( − 1 ) = 1 + 4 = 5
So, 1 + 2 i 1 = 5 1 − 2 i = 5 1 − 5 2 i .
Now, we need to square this result: ( 5 1 − 5 2 i ) 2 = ( 5 1 ) 2 − 2 × 5 1 × 5 2 i + ( 5 2 i ) 2
This simplifies to: 25 1 − 25 4 i + 25 4 ( − 1 )
The imaginary unit squared i 2 is − 1 , so: 25 1 − 25 4 i − 25 4
Combine like terms: ( 25 1 − 25 4 ) − 25 4 i = − 25 3 − 25 4 i
Hence, the real part of ( 1 + 2 i ) − 2 is − 25 3 , and the imaginary part is − 25 4 .
