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$0$
Calculate the value
$1 + \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ i } } } + \dfrac { 1 } { i ^ { 2 } } + \dfrac { 1 } { i ^ { 3 } }$
$ $ Calculate the rationalization of the complex number $ $
$1 + \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } \color{#FF6800}{ i } } { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } } + \dfrac { 1 } { i ^ { 2 } } + \dfrac { 1 } { i ^ { 3 } }$
$1 + \dfrac { \color{#FF6800}{ 1 } i } { - 1 } + \dfrac { 1 } { i ^ { 2 } } + \dfrac { 1 } { i ^ { 3 } }$
$ $ Multiplying any number by 1 does not change the value $ $
$1 + \dfrac { i } { - 1 } + \dfrac { 1 } { i ^ { 2 } } + \dfrac { 1 } { i ^ { 3 } }$
$1 + \dfrac { i } { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } + \dfrac { 1 } { i ^ { 2 } } + \dfrac { 1 } { i ^ { 3 } }$
$ $ Move the minus sign to the front of the fraction $ $
$1 \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ i } } { \color{#FF6800}{ 1 } } } + \dfrac { 1 } { i ^ { 2 } } + \dfrac { 1 } { i ^ { 3 } }$
$1 - \dfrac { i } { 1 } + \dfrac { 1 } { \color{#FF6800}{ i } ^ { \color{#FF6800}{ 2 } } } + \dfrac { 1 } { i ^ { 3 } }$
$ $ It is $ i^2 = -1$
$1 - \dfrac { i } { 1 } + \dfrac { 1 } { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } + \dfrac { 1 } { i ^ { 3 } }$
$1 - \dfrac { i } { 1 } + \dfrac { 1 } { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } + \dfrac { 1 } { i ^ { 3 } }$
$ $ Move the minus sign to the front of the fraction $ $
$1 - \dfrac { i } { 1 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 1 } } } + \dfrac { 1 } { i ^ { 3 } }$
$1 - \dfrac { i } { 1 } - \dfrac { 1 } { 1 } + \dfrac { 1 } { \color{#FF6800}{ i } ^ { \color{#FF6800}{ 3 } } }$
$i^2 $ is -1, so it is $ i^3=i^2\times i^1=-i$
$1 - \dfrac { i } { 1 } - \dfrac { 1 } { 1 } + \dfrac { 1 } { \color{#FF6800}{ - } \color{#FF6800}{ i } }$
$1 - \dfrac { i } { 1 } - \dfrac { 1 } { 1 } + \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ - } \color{#FF6800}{ i } } }$
$ $ Calculate the rationalization of the complex number $ $
$1 - \dfrac { i } { 1 } - \dfrac { 1 } { 1 } + \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } \color{#FF6800}{ i } } { \color{#FF6800}{ 1 } } }$
$1 - \dfrac { i } { 1 } - \dfrac { 1 } { 1 } + \dfrac { \color{#FF6800}{ 1 } i } { 1 }$
$ $ Multiplying any number by 1 does not change the value $ $
$1 - \dfrac { i } { 1 } - \dfrac { 1 } { 1 } + \dfrac { i } { 1 }$
$1 \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ i } } { \color{#FF6800}{ 1 } } } - \dfrac { 1 } { 1 } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \color{#FF6800}{ i } } { \color{#FF6800}{ 1 } } }$
$ $ Remove the two numbers if the values are the same and the signs are different $ $
$1 - \dfrac { 1 } { 1 }$
$\color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 1 } } }$
$ $ Subtract $ \dfrac { 1 } { 1 } $ from $ 1$
$\color{#FF6800}{ 0 }$
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