Divide 4/(9+i)

Problem

4/(9+i)

Solution

1. Identify the goal, which is to rationalize the denominator by removing the imaginary unit i

2. Find the conjugate of the denominator 9+i which is 9−i

3. Multiply both the numerator and the denominator by this conjugate.

4/(9+i)⋅(9−i)/(9−i)

4. Distribute the numerator.

4*(9−i)=36−4*i

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

(9+i)*(9−i)=9−i2

6. Substitute i2=−1 into the denominator.

81−(−1)=81+1=82

7. Combine the results into a single fraction.

(36−4*i)/82

8. Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 2

(18−2*i)/41

9. Split the expression into its real and imaginary parts.

18/41−2/41*i

Final Answer

4/(9+i)=18/41−2/41*i

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