Simplify the complex number. Express the answer in a+bi form. Use fractions in the answer.

Simplify the following:
(1+3 i)/(2-5 i)

 

Multiply numerator and denominator of (1+3 i)/(2-5 i) by 2+5 i:
((1+3 i) (2+5 i))/((2-5 i) (2+5 i))

 

(2-5 i) (2+5 i) = 2×2+2×5 i+-5 i×2+-5 i×5 i = 4+10 i-10 i+25 = 29:
((1+3 i) (2+5 i))/29

 

(1+3 i) (2+5 i) = 1×2+1×5 i+3 i×2+3 i×5 i = 2+5 i+6 i-15 = -13+11 i:
Answer: |-13 + 11i / 29

\(\dfrac{1+3i}{2-5i}\\ =\dfrac{(1+3i)(2+5i)}{(2-5i)(2+5i)}\\ =\dfrac{-13+11i}{29}\\ =-\dfrac{13}{29}+\dfrac{11i}{29}\)

.