Simplify (1+i)/i-3/(4-i)

Problem

(1+i)/i−3/(4−i)

Solution

1. Rationalize the first term by multiplying the numerator and denominator by −i (the conjugate of i.

((1+i)*(−i))/(i*(−i))

(−i−i2)/(−i2)

2. Simplify the first term using the identity i2=−1

(−i−(−1))/(−(−1))

(1−i)/1=1−i

3. Rationalize the second term by multiplying the numerator and denominator by the conjugate 4+i

(3*(4+i))/((4−i)*(4+i))

(12+3*i)/(16−i2)

4. Simplify the second term using the identity i2=−1

(12+3*i)/(16−(−1))

(12+3*i)/17

5. Combine the results by subtracting the simplified second term from the simplified first term.

(1−i)−(12+3*i)/17

6. Find a common denominator to perform the subtraction.

(17*(1−i))/17−(12+3*i)/17

(17−17*i−12−3*i)/17

7. Combine like terms in the numerator.

(5−20*i)/17

Final Answer

(1+i)/i−3/(4−i)=5/17−20/17*i

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