Write in Standard Form 6/(5+i)

Problem

6/(5+i)

Solution

1. Identify the conjugate of the denominator. The denominator is 5+i so its complex conjugate is 5−i

2. Multiply both the numerator and the denominator by this conjugate to eliminate the imaginary unit from the denominator.

6/(5+i)⋅(5−i)/(5−i)

3. Distribute the numerator.

6*(5−i)=30−6*i

4. Expand the denominator using the difference of squares property (a+b*i)*(a−b*i)=a2+b2

(5+i)*(5−i)=5−i2

25−(−1)=26

5. Combine the results into a single fraction.

(30−6*i)/26

6. Simplify the fraction by dividing each term by the greatest common divisor, which is 2

30/26−(6*i)/26

15/13−3/13*i

Final Answer

6/(5+i)=15/13−3/13*i

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