simplify 1/(x+iy)

Solution

simplify

\frac{1}{x + i y}

\frac{x}{x ^{2} + y ^{2}} - \frac{y}{x ^{2} + y ^{2}} i

Solution steps

\frac{1}{x + i y}

Rewrite

x + i y

in standard complex form:

x + y i

= \frac{1}{x + y i}

Multiply by the conjugate

\frac{x - y i}{x - y i}

= \frac{1 \cdot ( x - y i )}{( x + y i ) ( x - y i )}

Simplify

\frac{1 \cdot ( x - y i )}{( x + y i ) ( x - y i )} : \frac{x - y i}{x ^{2} + y ^{2}}

= \frac{x - y i}{x ^{2} + y ^{2}}

Rewrite in standard complex form:

\frac{x}{x ^{2} + y ^{2}} - \frac{y}{x ^{2} + y ^{2}} i

= \frac{x}{x ^{2} + y ^{2}} - \frac{y}{x ^{2} + y ^{2}} i

(3 b - a) \cdot 4 c + 3 b - a

2 (r + 3) = 8

- 4 x \geq 32

\frac{7}{3} - \frac{5}{2}

4 x^{2} - 3 x - 12 = 0