Question

me ajudem pfv
1) A expressão i¹³+ i¹⁵ é igual a:

a) 0

b) i

c) – i

d) – 2i

e) 3i



2) Calcule as seguintes potências de i

a) i³⁵



b) i⁹⁴



c)i⁴²

d)i⁴⁵

e)i¹⁰⁸¹

f)i¹⁶⁰¹

3)calcule o valor das expressoes,sendo i a unidade imaginária:
a)i¹⁰⁰-i²⁰⁰
b)i³⁶+i¹⁰²
c)i¹²³+i¹⁸⁰

​

Answer 1

1 )

i¹³ + i¹⁵ =

(i²)⁶.i + (i²)⁷.i

(-1)⁶.i + (-1)⁷.i

(1).i + (-1).i

i - i

0

2)

a) i³⁵ = (i²)¹⁷.i¹ = (-1)¹⁷. i = -1. i = -i

b) i⁹⁴ = (i²)⁴⁷ = (-1)⁴⁷ = -1

c)i⁴² = (i²)²¹ = (-1)²¹ = -1

d)i⁴⁵ = (i²)²².i¹ = (-1)²². i = (1). i = i

e)i¹⁰⁸¹ = (i²)⁵⁴⁰. i¹ = (-1)⁵⁴⁰. i = 1. i = i

f)i¹⁶⁰¹ = (i²)⁸⁰⁰. i¹ = (-1)⁸⁰⁰. i = (1). i = i

3)

a)i¹⁰⁰-i²⁰⁰ = (i⁴)²⁵ - (i⁴)⁵⁰ = (1) - (1) = 0

b)i³⁶+i¹⁰² = (i²)¹⁸+(i²)⁵¹ =(-1)¹⁸+(-1)⁵¹ =1-1=0

c)i¹²³+i¹⁸⁰= (i²)⁶¹.i +(i²)⁹⁰= -1.i + 1 = -i +1

Answer 2

1)

$\mathsf{i^{13}=i^1=i}\\\mathsf{i^{15}=i^{3}=-i}\\\mathsf{i^{13}+i^{15}=i-i=0}$

$\dotfill$

2)

a) i³⁵=i³=-i

b) i⁹⁴=i²=-1

c)i⁴²=i²=-1

d)i⁴⁵=i¹=i

e)i¹⁰⁸¹=i¹=i

f)i¹⁶⁰¹=i¹=i

$\dotfill$

3)

a)

$\mathsf{i^{100}=i^0=1}\\\mathsf{i^{200}=i^0=1}\\\mathsf{i^{100}-i^{200}=1-1=0}$

b)

$\mathsf{i^{36}=i^0=1}\\\mathsf{i^{102}=i^2=-1}\\\mathsf{i^{36}+i^{102}=1-1=0}$

c)

$\mathsf{i^{123}=i^3=-i}\\\mathsf{i^{180}=i^0=1}\\\mathsf{i^{123}+i^{180}=-i+1}$