What is (3i^{171}+2)/(4i-i^{20)} ?

The solution to (3i^{171}+2)/(4i-i^{20)} is -14/17-i 5/17

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simplify (3i^{171}+2)/(4i-i^{20)}

simplify

\frac{3 i ^{171} + 2}{4 i - i ^{20}}

- \frac{14}{17} - i \frac{5}{17}

Solution steps

\frac{3 i ^{171} + 2}{4 i - i ^{20}}

3 i^{171} + 2 = - 3 i + 2

= \frac{- 3 i + 2}{4 i - i ^{20}}

4 i - i^{20} = 4 i - 1

= \frac{- 3 i + 2}{4 i - 1}

Apply complex arithmetic rule:

\frac{a + bi}{c + d i} = \frac{( c - d i ) ( a + bi )}{( c - d i ) ( c + d i )} = \frac{( a c + b d ) + ( b c - a d ) i}{c ^{2} + d ^{2}}

a = 2, b = - 3, c = - 1, d = 4

= \frac{( 2 ( - 1 ) + ( - 3 ) \cdot 4 ) + ( - 3 ( - 1 ) - 2 \cdot 4 ) i}{( - 1 ) ^{2} + 4 ^{2}}

Refine

= \frac{- 14 - 5 i}{17}

Rewrite

\frac{- 14 - 5 i}{17}

in standard complex form:

- \frac{14}{17} - \frac{5}{17} i

= - \frac{14}{17} - \frac{5}{17} i

\frac{36}{- 3}

\frac{24}{42} \frac{18}{42}

(2 + 5 i) (3 - 4 i)

3 (6 v + 3)

9 (a + x)^{2} - 12 (a + x) + 4

What is (3i^{171}+2)/(4i-i^{20)} ?
The solution to (3i^{171}+2)/(4i-i^{20)} is -14/17-i 5/17