What is the answer to 1/z =(2+8i)/(8i) ?

The answer to 1/z =(2+8i)/(8i) is z= 16/17+4/17 i

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1/z =(2+8i)/(8i)

\frac{1}{z} = \frac{2 + 8 i}{8 i}

z = \frac{16}{17} + \frac{4}{17} i

Solution steps

\frac{1}{z} = \frac{2 + 8 i}{8 i}

Substitute

z = x + y i

\frac{1}{x + y i} = \frac{2 + 8 i}{8 i}

Apply fraction cross multiply: if

\frac{a}{b} = \frac{c}{d}

then

a \cdot d = b \cdot c

1 \cdot 8 i = (x + y i) (2 + 8 i)

Expand

8 i = (2 x - 8 y) + i (2 y + 8 x)

Rewrite

8 i

in standard complex form:

0 + 8 i

0 + 8 i = (2 x - 8 y) + i (2 y + 8 x)

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[0 = 2 x - 8 y 8 = 2 y + 8 x]

[0 = 2 x - 8 y 8 = 2 y + 8 x] : x = \frac{16}{17}, y = \frac{4}{17}

Substitute back

z = x + y i

z = \frac{16}{17} + \frac{4}{17} i

z^{2} = - 3 + 4 i

(1 + ai) (1 + bi) = b + 9 i - a

(3 x - i) (2 y + i) = - 11 + 7 i

4 + x + 3 i = 8 - (y + 1) i

- \frac{1}{z - i} + 1 - i - z = 0

What is the answer to 1/z =(2+8i)/(8i) ?
The answer to 1/z =(2+8i)/(8i) is z= 16/17+4/17 i