Simplify ((3 + i))/((5 + 2i))

Question

Simplify

`((3 + i))/((5 + 2i))`

Answer 1

-
`(1)/(29)` i +
`(17)/(29)`

Explanation:

to simplify the fraction ewe require to rationalise the denominator.

This is done by multiplying the numerator/ denominator by the conjugate of the denominator.

the conjugate of 5 + 2i is 5 - 2i, hence

=
`((3+i)(5-2i))/((5+2i)(5-2i))`

=
`(15-i-2i^2)/(25-4i^2)`

[ i² = (
`√(-1)`)² = - 1 ]

=
`(-i+17)/(29)` = -
`(1)/(29)` i +
`(17)/(29)`

Answer 2

17-i

--------

29

Explanation:

To simply fractions with complex numbers, we have to multiply by the complex conjugate of the denominator.

The denominator is 5+2i, so the complex conjuate is 5-2i

3+i 5-2i

------ * ---------------

5+2i 5-2i

Simplifying the numerator

(3+i) * (5-2i) = 3*5 + 5i -3*2i -2i^2 = 15 +5i -6i -2(-1) = 15-i+2 = 17-i

Simplifying the denominator

(5+2i) (5-2i) = 25 -10i + 10i -4i^2 = 25 -4(-1) = 29

17-i

--------

29