# Simplify with rational denominator: ((1+i)(2-i))/((2+i)*i)

Asked on by greg1234

lemjay | High School Teacher | (Level 2) Senior Educator

Posted on

(1+i)(2-i)

---------                      Use distributive property to simplify.

(2+i)i

2 - i + 2i - i^2

= ---------------           Note that i^2 = -1. So, replace i^2 with -1.

2i + i^2

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

= ----------------  =    ----------  =   --------

2i + (-1)                2i - 1             2i - 1

3 + i      2i + 1

= ------- * -------       Rationalize by multiplying the top and

2i - 1      2i + 1        bottom by the conjugate of 2i - 1.

6i + 3 + 2i^2 +  i            7i + 3 + 2i^2           7i + 3 + 2(-1)

= -------------------   =    ----------------  =    ---------------

4i^2 + 2i - 2i - 1             4i^2  - 1                  4(-1) - 1

7i  + 1

= -----------

- 5

Answer:  (7i + 1) / -5

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The expression `((1+i)(2-i))/((2+i)*i)` has to be simplified and the denominator converted to a rational number.

`((1+i)(2-i))/((2+i)*i)`

=> `((1+i)(2-i)(2 - i)i)/((2+i)*i*(2 - i)*i)`

=> `((1+i)(2-i)(2 - i)i)/((4 - i^2)*i^2)`

=> `-((i+i^2)(2-i)^2)/5`

=> `-((i - 1)(4 + i^2 -4i))/5`

=> `-((i - 1)(4 -1 - 4i))/5`

=> `-((i - 1)(3 - 4i))/5`

=> `-(3i - 4i^2 - 3 + 4i)/5`

=> `-(1 + 7i)/5`

The expression `((1+i)(2-i))/((2+i)*i) = -(1 + 7i)/5`

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