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Popular Algebra >

simplify (1+i)^{20}

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Solution

simplify (1+i)20

Solution

−1024
Solution steps
(1+i)20
Apply binomial theorem: (a+b)n=i=0∑n​(in​)a(n−i)bia=1,b=i
=i=0∑20​(i20​)⋅1(20−i)ii
Expand summation
=0!(20−0)!20!​⋅120i0+1!(20−1)!20!​⋅119i1+2!(20−2)!20!​⋅118i2+3!(20−3)!20!​⋅117i3+4!(20−4)!20!​⋅116i4+5!(20−5)!20!​⋅115i5+6!(20−6)!20!​⋅114i6+7!(20−7)!20!​⋅113i7+8!(20−8)!20!​⋅112i8+9!(20−9)!20!​⋅111i9+10!(20−10)!20!​⋅110i10+11!(20−11)!20!​⋅19i11+12!(20−12)!20!​⋅18i12+13!(20−13)!20!​⋅17i13+14!(20−14)!20!​⋅16i14+15!(20−15)!20!​⋅15i15+16!(20−16)!20!​⋅14i16+17!(20−17)!20!​⋅13i17+18!(20−18)!20!​⋅12i18+19!(20−19)!20!​⋅11i19+20!(20−20)!20!​⋅10i20
Simplify 0!(20−0)!20!​⋅120i0:1
Simplify 1!(20−1)!20!​⋅119i1:20i
Simplify 2!(20−2)!20!​⋅118i2:190i2
Simplify 3!(20−3)!20!​⋅117i3:1140i3
Simplify 4!(20−4)!20!​⋅116i4:4845i4
Simplify 5!(20−5)!20!​⋅115i5:15504i5
Simplify 6!(20−6)!20!​⋅114i6:38760i6
Simplify 7!(20−7)!20!​⋅113i7:77520i7
Simplify 8!(20−8)!20!​⋅112i8:125970i8
Simplify 9!(20−9)!20!​⋅111i9:167960i9
Simplify 10!(20−10)!20!​⋅110i10:184756i10
Simplify 11!(20−11)!20!​⋅19i11:167960i11
Simplify 12!(20−12)!20!​⋅18i12:125970i12
Simplify 13!(20−13)!20!​⋅17i13:77520i13
Simplify 14!(20−14)!20!​⋅16i14:38760i14
Simplify 15!(20−15)!20!​⋅15i15:15504i15
Simplify 16!(20−16)!20!​⋅14i16:4845i16
Simplify 17!(20−17)!20!​⋅13i17:1140i17
Simplify 18!(20−18)!20!​⋅12i18:190i18
Simplify 19!(20−19)!20!​⋅11i19:20i19
Simplify 20!(20−20)!20!​⋅10i20:i20
=1+20i+190i2+1140i3+4845i4+15504i5+38760i6+77520i7+125970i8+167960i9+184756i10+167960i11+125970i12+77520i13+38760i14+15504i15+4845i16+1140i17+190i18+20i19+i20
Simplify 1+20i+190i2+1140i3+4845i4+15504i5+38760i6+77520i7+125970i8+167960i9+184756i10+167960i11+125970i12+77520i13+38760i14+15504i15+4845i16+1140i17+190i18+20i19+i20:−1024
=−1024

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Frequently Asked Questions (FAQ)

  • What is (1+i)^{20} ?

    The solution to (1+i)^{20} is -1024

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