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

The solution to (4i-i^{20})/(3i^{171)+2} is -14/13+i 5/13

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

simplify

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

- \frac{14}{13} + i \frac{5}{13}

Solution steps

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

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

= \frac{4 i - 1}{3 i ^{171} + 2}

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

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

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

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

Refine

= \frac{- 14 + 5 i}{13}

Rewrite

\frac{- 14 + 5 i}{13}

in standard complex form:

- \frac{14}{13} + \frac{5}{13} i

= - \frac{14}{13} + \frac{5}{13} i

\frac{35 ab}{- 49 a ^{4} b ^{3}}

(1 - i)^{- 1}

\frac{- 56}{- 7}

28 + 8

\frac{3 ( a - b )}{9} + \frac{4}{9} b

What is (4i-i^{20})/(3i^{171)+2} ?
The solution to (4i-i^{20})/(3i^{171)+2} is -14/13+i 5/13