perform the indicated operation

Question

1+i/4-5i

 

Express the answer in form a+bi. The dash is because it divides the two equations

Answer 1

I think you are trying to solve:  (1+i)/(4-5i)

You really need to use parentheses to denote what are individual quantities.

If so then the following is the procedure to use.

 

In order to clear the 'i' from the denominator, (4-5i) should be multiplied by its conjugate (4+5i)

Which means we multiply the equation by (4+5i)/(4+5i)

 

(1+i)/(4-5i) = [(1+i)(4+5i)]/[(4-5i)(4+5i)]  = (4+5i+4i+5i²)/(16 +20i-20i-25i²)

 

= (4+9i-5)/(16-25(-1)) = (-1 + 9i)/(16 + 25) = (-1+9i)/41 = -1/41 + (9/41)i

Answer 2

(1 + i)/(4 - 5i)               This is a more precise way of writing the fraction

 

Rationalize the denominator by multiplying by the complex conjugate.

 

(1 + i)(4 + 5i) / (4 - 5i)(4 + 5i)

(4 + 4i + 5i + 5i²) / (16 - 20i + 20i + 25i²)

(4 + 9i - 5) / (16 + 25)

(-1 + 9i) /  41

-1/41 - (9/41)i