Question

Can You please help me work through this step by step?
Simplify the given expression.

2 divided by the quantity of 2 plus 5i

Answer 1

do you recall what a "conjugate" is?

Answer 2

is 5i pi?

Answer 3

Does it look like this?
\[\frac{ 2 }{ 2 + 5i }\]

Answer 4

so... anyhow.. the conjugate will be the same pair, different sign in between
so for a+b conjugate is a-b
for -a+b conjugate is -a-b
for say a -b conjugate will be a+b
for -a -b conjugate will be -a +b

so "simplifying" a complex rational, means getting rid of that pesky "i" at the bottom
and you'd do it by using the conjugate of the denominator and multiplying the fraction by it, that is \(\bf \cfrac{2}{2+5i}\cdot \cfrac{{\color{blue}{ 2-5i}}}{{\color{blue}{ 2-5i}} }\implies \cfrac{2( 2-5i)}{(2+5i)(2-5i)}\\ \quad \\
recall\implies {\color{blue}{ (a-b)(a+b) = a^2-b^2\qquad and\qquad i^2=-1}}\\ \quad \\

\cfrac{2( 2-5i)}{(2+5i)(2-5i)}\implies \cfrac{2( 2-5i)}{2^2-(5i)^2}\implies \cfrac{2( 2-5i)}{2^2-(5^2i^2)}\\ \quad \\
\cfrac{2( 2-5i)}{2^2-(25\cdot -1)}\)

Answer 5

expand the numerator, and simplify away

Answer 6

oh my