Simplify $\displaystyle(1-i)/(2+3i)$, where $i^2 = -1.$

Question

Simplify
`$\displaystyle(1-i)/(2+3i)$`, where
`$i^2 = -1.$`

Answer 1

(-1-5i)/13

Explanation:

You can't leave that i on the bottom of the fraction. Thats why it needs work. MULTIPLY the top and the bottom by the CONJUGATE of the bottom term. That means look at the bottom, copy those numbers but make the sign the opposite.

Your question:

(1-i)/(2+3i)

CONJUGATE:

2 - 3i

MULTIPY:

(1-i)/(2+3i) × (2-3i)/(2-3i)

You multiply fractions by multiplying top×top and bottom ×bottom, just straight across.

On top:

(1-i)(2-3i)

2-3i-2i+3i^2

2-3i-2i-3 Note: i^2=-1

-1-5i This goes on top.

On bottom:

(2+3i)(2-3i)

4-6i+6i-9i^2

4-9i^2

4+9

13

final answer:

(-1-5i)/13