Algebra 2

Algebra Level pending

Evaluate

( i + 1 ) 20 { (i\quad +\quad 1) }^{ 20 }

1 , 048 , 576 i 1,048,576i 1024 i 1024i − 1024 -1024

Dominick Hing by Dominick Hing , 23, USA

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2 solutions

展豪 張
Mar 13, 2016

( i + 1 ) 20 = ( 2 e i π 4 ) 20 = 2 10 e i 5 π = 1024 e i π = − 1024 \displaystyle(i+1)^{20}=(\sqrt2e^{i\frac\pi 4})^{20}=2^{10}e^{i5\pi}=1024e^{i\pi}=-1024

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Dominick Hing
Sep 28, 2014

Rewrite as

[ ( i + 1 ) 2 ] 10 [{ { (i\quad +\quad 1) }^{ 2 }] }^{ 10 }

Since

( i + 1 ) 2 = 2 i { (i+1) }^{ 2 }=2i

then

[ ( i + 1 ) 2 ] 10 = ( 2 i ) 10 = 1024 i 10 = − 1024 [{ { (i\quad +\quad 1) }^{ 2 }] }^{ 10 }={ (2i) }^{ 10 }=1024{ i }^{ 10 }=-1024

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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