Question

Determine o número natural n tal que (2i)^n + (1+i)^2n = 32

Answer 1

Explicação passo-a-passo:

$(2i {)}^{n} + (1 + i {)}^{2n} = 32$

$(2i {)}^{n} +( (1 + i {)}^{2} {)}^{n} = 32$

$(2i {)}^{n} + (1 + 2i + {i}^{2} {)}^{n} = 32$

Sendo:

${i}^{2} = - 1$

$(2i {)}^{n} + (1 + 2i - 1 {)}^{n} = 32$

$(2i {)}^{n} + (2i {)}^{n} = 32$

$2(2i {)}^{n} = 32$

$(2i {)}^{n} = 16$

$(2 \sqrt{( - 1)} {)}^{n} = 16$

${2}^{n} ( - 1 {)}^{ \frac{n}{2} } = 16$

Observe que n precisa ser par.

${2}^{n} = 16$

${2}^{n} = {2}^{4}$

$n = 4$