Question

Simplify
$\frac{(3 + i)}{(5 + 2i)}$

Answer 1

Answer:

17-i

--------

29

Step-by-step 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

Answer 2

Answer:

- $\frac{1}{29}$ i + $\frac{17}{29}$

Step-by-step 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

= $\frac{(3+i)(5-2i)}{(5+2i)(5-2i)}$

= $\frac{15-i-2i^2}{25-4i^2}$

[ i² = ($\sqrt{-1}$)² = - 1 ]

= $\frac{-i+17}{29}$ = - $\frac{1}{29}$ i + $\frac{17}{29}$