Question

Desenvolva cada expressão abaixo

Answer 1

Explicação passo-a-passo:

9)

Veja que:

• $\sf i^2=-1$

• $\sf i^3=i^2\cdot i$

$\sf i^3=(-1)\cdot i$

$\sf i^3=-i$

• $\sf i^4=(i^2)^2$

$\sf i^4=(-1)^2$

$\sf i^4=1$

• $\sf i+i^2+i^3+i^4=i-1-i+1$

$\sf i+i^2+i^3+i^4=0$

A cada 4 potências consecutivas de i, obtemos soma zero

Assim:

• $\sf i+i^2+i^3+i^4=0$

• $\sf i^5+i^6+i^7+i^8=0$

• $\sf i^9+i^{10}+i^{11}+i^{12}=0$

...

• $\sf i^{37}+i^{38}+i^{39}+i^{40}=0$

Logo:

$\sf A=i+i^2+i^3+\dots+i^{41}$

$\sf A=0+0+0+\dots+0+i^{41}$

$\sf A=i^{41}$

$\sf A=i^{40}\cdot i$

$\sf A=(i^2)^{20}\cdot i$

$\sf A=(-1)^{20}\cdot i$

$\sf A=1\cdot i$

$\sf \red{A=i}$

b)

$\sf A=i^3+i^4+i^5+\dots+i^{26}$

Somando $\sf i+i^2$ nos dois lados dessa igualdade:

$\sf A+i+i^2=i+i^2+i^3+i^4+i^5+\dots+i^{26}$

Temos:

• $\sf i+i^2+i^3+i^4=0$

• $\sf i^5+i^6+i^7+i^8=0$

• $\sf i^9+i^{10}+i^{11}+i^{12}=0$

...

• $\sf i^{21}+i^{22}+i^{23}+i^{24}=0$

Assim:

$\sf A+i+i^2=i+i^2+i^3+i^4+i^5+\dots+i^{26}$

$\sf A+i-1=0+0+0\dots+i^{25}+i^{26}$

$\sf A+i-1=i^{25}+i^{26}$

• $\sf i^{25}=i^{24}\cdot i$

$\sf i^{25}=(i^2)^{12}\cdot i$

$\sf i^{25}=(-1)^{12}\cdot i$

$\sf i^{25}=1\cdot i$

$\sf i^{25}=i$

• $\sf i^{26}=i^{25}\cdot i$

$\sf i^{26}=i\cdot i$

$\sf i^{26}=i^2$

$\sf i^{26}=-1$

Logo:

$\sf A+i-1=i^{25}+i^{26}$

$\sf A+i-1=i-1$

$\sf A=i-1-i+1$

$\sf \red{A=0}$

c)

$\sf (-2i)^7=(-2)^7\cdot i^7$

$\sf (-2i)^7=(-128)\cdot i^6\cdot i$

$\sf (-2i)^7=(-128)\cdot(i^2)^3\cdot i$

$\sf (-2i)^7=(-128)\cdot(-1)^3\cdot i$

$\sf (-2i)^7=(-128)\cdot(-1)\cdot i$

$\sf (-2i)^7=(-128)\cdot(-i)$

$\sf \red{(-2i)^7=128i}$

d)

• $\sf i^{121}=i^{120}\cdot i$

$\sf i^{121}=(i^2)^{60}\cdot i$

$\sf i^{121}=(-1)^{60}\cdot i$

$\sf i^{121}=1\cdot i$

$\sf i^{121}=i$

• $\sf i^{20}=(i^2)^{10}$

$\sf i^{20}=(-1)^{10}$

$\sf i^{20}=1$

• $\sf i^{18}=(i^2)^{9}$

$\sf i^{18}=(-1)^{9}$

$\sf i^{18}=-1$

Assim:

$\sf A=\dfrac{i^{121}-i^{20}}{i^{18}}$

$\sf A=\dfrac{i-1}{-1}$

$\sf A=-i+1$

$\sf \red{A=1-i}$