Write in Standard Form (5+i)/(5-i)

Problem

(5+i)/(5−i)

Solution

1. Identify the conjugate of the denominator. To remove the imaginary unit from the denominator, we must multiply the numerator and the denominator by the complex conjugate of 5−i which is 5+i

2. Multiply the numerator and the denominator by the conjugate.

(5+i)/(5−i)⋅(5+i)/(5+i)

3. Expand the numerator using the FOIL method (First, Outer, Inner, Last).

(5+i)*(5+i)=25+5*i+5*i+i2

25+10*i+i2

4. Expand the denominator using the difference of squares pattern.

(5−i)*(5+i)=5−i2

25−i2

5. Substitute −1 for i2 in both the numerator and the denominator.

Numerator: *25+10*i+(−1)=24+10*i

Denominator: *25−(−1)=26

6. Simplify the resulting fraction by dividing both terms in the numerator by the denominator.

(24+10*i)/26

24/26+(10*i)/26

7. Reduce the fractions to their simplest form.

12/13+5/13*i

Final Answer

(5+i)/(5−i)=12/13+5/13*i

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