Simplify. [tex]\(\frac{-4i}{1 + 3i} = -\frac{?}{?} - \frac{?}{?}i\)[/tex]
Answer (2)
To simplify 1 + 3 i − 4 i , we multiply by the conjugate of the denominator, resulting in − 5 6 − 5 2 i . This gives the answers for the question marks as 6, 5, 2, and 5 respectively.
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To simplify the expression 1 + 3 i − 4 i into the form − ? ? − ? ? i , we need to eliminate the imaginary part from the denominator using a process called 'rationalizing the denominator'.
Step-by-Step Solution:
Multiply the Numerator and Denominator by the Conjugate of the Denominator :
The conjugate of 1 + 3 i is 1 − 3 i . To rationalize the denominator, multiply both the numerator and the denominator by this conjugate:
1 + 3 i − 4 i × 1 − 3 i 1 − 3 i = ( 1 + 3 i ) ( 1 − 3 i ) − 4 i ( 1 − 3 i )
Simplify the Denominator :
Use the formula for the difference of squares, ( a + b ) ( a − b ) = a 2 − b 2 :
( 1 + 3 i ) ( 1 − 3 i ) = 1 2 − ( 3 i ) 2 = 1 − 9 i 2
Since i 2 = − 1 , it follows that:
1 − 9 ( − 1 ) = 1 + 9 = 10
Simplify the Numerator :
Distribute − 4 i across 1 − 3 i :
− 4 i ( 1 − 3 i ) = − 4 i + 12 i 2
Again, since i 2 = − 1 :
− 4 i + 12 ( − 1 ) = − 4 i − 12
Combine and Simplify into Real and Imaginary Parts :
Combine the real and imaginary parts:
10 − 4 i − 12 = 10 − 12 + 10 − 4 i
Simplifying both fractions:
− 10 12 = − 5 6 , 10 − 4 i = − 5 2 i
Thus, the simplified expression is:
− 5 6 − 5 2 i
So, 1 + 3 i − 4 i simplifies to − 5 6 − 5 2 i .
