Cálculo Exemplos

${(3 (\cos (30 °) + i \sin (30 °)))}^{6}$

Etapa 1

Simplifique os termos.

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Etapa 1.1

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Etapa 1.1.1

O valor exato de $\cos (30 °)$ é $\frac{\sqrt{3}}{2}$ .

${(3 (\frac{\sqrt{3}}{2} + i \sin (30 °)))}^{6}$

Etapa 1.1.2

O valor exato de $\sin (30 °)$ é $\frac{1}{2}$ .

${(3 (\frac{\sqrt{3}}{2} + i \frac{1}{2}))}^{6}$

Etapa 1.1.3

Combine $i$ e $\frac{1}{2}$ .

${(3 (\frac{\sqrt{3}}{2} + \frac{i}{2}))}^{6}$

Etapa 1.2

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Etapa 1.2.1

Aplique a propriedade distributiva.

${(3 \frac{\sqrt{3}}{2} + 3 \frac{i}{2})}^{6}$

Etapa 1.2.2

Combine $3$ e $\frac{\sqrt{3}}{2}$ .

${(\frac{3 \sqrt{3}}{2} + 3 \frac{i}{2})}^{6}$

Etapa 1.2.3

Combine $3$ e $\frac{i}{2}$ .

${(\frac{3 \sqrt{3}}{2} + \frac{3 i}{2})}^{6}$

Etapa 2

Use o teorema binomial.

${(\frac{3 \sqrt{3}}{2})}^{6} + 6 {(\frac{3 \sqrt{3}}{2})}^{5} \frac{3 i}{2} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3

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Etapa 3.1

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Etapa 3.1.1

Use a regra da multiplicação de potências ${(a b)}^{n} = a^{n} b^{n}$ para distribuir o expoente.

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Etapa 3.1.1.1

Aplique a regra do produto a $\frac{3 \sqrt{3}}{2}$ .

$\frac{{(3 \sqrt{3})}^{6}}{2^{6}} + 6 {(\frac{3 \sqrt{3}}{2})}^{5} \frac{3 i}{2} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.1.2

Aplique a regra do produto a $3 \sqrt{3}$ .

$\frac{3^{6} {\sqrt{3}}^{6}}{2^{6}} + 6 {(\frac{3 \sqrt{3}}{2})}^{5} \frac{3 i}{2} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.2

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Etapa 3.1.2.1

Eleve $3$ à potência de $6$ .

$\frac{729 {\sqrt{3}}^{6}}{2^{6}} + 6 {(\frac{3 \sqrt{3}}{2})}^{5} \frac{3 i}{2} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.2.2

Reescreva ${\sqrt{3}}^{6}$ como $3^{3}$ .

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Etapa 3.1.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescrever $\sqrt{3}$ como $3^{\frac{1}{2}}$ .

$\frac{729 {(3^{\frac{1}{2}})}^{6}}{2^{6}} + 6 {(\frac{3 \sqrt{3}}{2})}^{5} \frac{3 i}{2} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.2.2.2

Aplique a regra da multiplicação de potências e multiplique os expoentes, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{729 \cdot 3^{\frac{1}{2} \cdot 6}}{2^{6}} + 6 {(\frac{3 \sqrt{3}}{2})}^{5} \frac{3 i}{2} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.2.2.3

Combine $\frac{1}{2}$ e $6$ .

$\frac{729 \cdot 3^{\frac{6}{2}}}{2^{6}} + 6 {(\frac{3 \sqrt{3}}{2})}^{5} \frac{3 i}{2} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.2.2.4

Cancele o fator comum de $6$ e $2$ .

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Etapa 3.1.2.2.4.1

Fatore $2$ de $6$ .

$\frac{729 \cdot 3^{\frac{2 \cdot 3}{2}}}{2^{6}} + 6 {(\frac{3 \sqrt{3}}{2})}^{5} \frac{3 i}{2} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.2.2.4.2

Cancele os fatores comuns.

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Etapa 3.1.2.2.4.2.1

Fatore $2$ de $2$ .

$\frac{729 \cdot 3^{\frac{2 \cdot 3}{2 (1)}}}{2^{6}} + 6 {(\frac{3 \sqrt{3}}{2})}^{5} \frac{3 i}{2} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.2.2.4.2.2

Cancele o fator comum.

$\frac{729 \cdot 3^{\frac{2 \cdot 3}{2 \cdot 1}}}{2^{6}} + 6 {(\frac{3 \sqrt{3}}{2})}^{5} \frac{3 i}{2} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.2.2.4.2.3

Reescreva a expressão.

$\frac{729 \cdot 3^{\frac{3}{1}}}{2^{6}} + 6 {(\frac{3 \sqrt{3}}{2})}^{5} \frac{3 i}{2} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.2.2.4.2.4

Divida $3$ por $1$ .

$\frac{729 \cdot 3^{3}}{2^{6}} + 6 {(\frac{3 \sqrt{3}}{2})}^{5} \frac{3 i}{2} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.2.3

Eleve $3$ à potência de $3$ .

$\frac{729 \cdot 27}{2^{6}} + 6 {(\frac{3 \sqrt{3}}{2})}^{5} \frac{3 i}{2} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.3

Eleve $2$ à potência de $6$ .

$\frac{729 \cdot 27}{64} + 6 {(\frac{3 \sqrt{3}}{2})}^{5} \frac{3 i}{2} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.4

Multiplique $729$ por $27$ .

$\frac{19683}{64} + 6 {(\frac{3 \sqrt{3}}{2})}^{5} \frac{3 i}{2} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.5

Cancele o fator comum de $2$ .

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Etapa 3.1.5.1

Fatore $2$ de $6 {(\frac{3 \sqrt{3}}{2})}^{5}$ .

$\frac{19683}{64} + 2 (3 {(\frac{3 \sqrt{3}}{2})}^{5}) \frac{3 i}{2} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.5.2

Cancele o fator comum.

Etapa 3.1.5.3

Reescreva a expressão.

$\frac{19683}{64} + 3 {(\frac{3 \sqrt{3}}{2})}^{5} (3 i) + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.6

Multiplique $3$ por $3$ .

$\frac{19683}{64} + 9 {(\frac{3 \sqrt{3}}{2})}^{5} i + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.7

Use a regra da multiplicação de potências ${(a b)}^{n} = a^{n} b^{n}$ para distribuir o expoente.

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Etapa 3.1.7.1

Aplique a regra do produto a $\frac{3 \sqrt{3}}{2}$ .

$\frac{19683}{64} + 9 \frac{{(3 \sqrt{3})}^{5}}{2^{5}} i + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.7.2

Aplique a regra do produto a $3 \sqrt{3}$ .

$\frac{19683}{64} + 9 \frac{3^{5} {\sqrt{3}}^{5}}{2^{5}} i + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.8

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Etapa 3.1.8.1

Eleve $3$ à potência de $5$ .

$\frac{19683}{64} + 9 \frac{243 {\sqrt{3}}^{5}}{2^{5}} i + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.8.2

Reescreva ${\sqrt{3}}^{5}$ como $\sqrt{3^{5}}$ .

$\frac{19683}{64} + 9 \frac{243 \sqrt{3^{5}}}{2^{5}} i + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.8.3

Eleve $3$ à potência de $5$ .

$\frac{19683}{64} + 9 \frac{243 \sqrt{243}}{2^{5}} i + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.8.4

Reescreva $243$ como $9^{2} \cdot 3$ .

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Etapa 3.1.8.4.1

Fatore $81$ de $243$ .

$\frac{19683}{64} + 9 \frac{243 \sqrt{81 (3)}}{2^{5}} i + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.8.4.2

Reescreva $81$ como $9^{2}$ .

$\frac{19683}{64} + 9 \frac{243 \sqrt{9^{2} \cdot 3}}{2^{5}} i + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.8.5

Elimine os termos abaixo do radical.

$\frac{19683}{64} + 9 \frac{243 \cdot 9 \sqrt{3}}{2^{5}} i + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.8.6

Multiplique $243$ por $9$ .

$\frac{19683}{64} + 9 \frac{2187 \sqrt{3}}{2^{5}} i + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.9

Eleve $2$ à potência de $5$ .

$\frac{19683}{64} + 9 \frac{2187 \sqrt{3}}{32} i + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.10

Multiplique $9 \frac{2187 \sqrt{3}}{32}$ .

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Etapa 3.1.10.1

Combine $9$ e $\frac{2187 \sqrt{3}}{32}$ .

$\frac{19683}{64} + \frac{9 (2187 \sqrt{3})}{32} i + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.10.2

Multiplique $2187$ por $9$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3}}{32} i + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.11

Combine $\frac{19683 \sqrt{3}}{32}$ e $i$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + 15 {(\frac{3 \sqrt{3}}{2})}^{4} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.12

Use a regra da multiplicação de potências ${(a b)}^{n} = a^{n} b^{n}$ para distribuir o expoente.

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Etapa 3.1.12.1

Aplique a regra do produto a $\frac{3 \sqrt{3}}{2}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + 15 \frac{{(3 \sqrt{3})}^{4}}{2^{4}} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.12.2

Aplique a regra do produto a $3 \sqrt{3}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + 15 \frac{3^{4} {\sqrt{3}}^{4}}{2^{4}} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.13

Simplifique o numerador.

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Etapa 3.1.13.1

Eleve $3$ à potência de $4$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + 15 \frac{81 {\sqrt{3}}^{4}}{2^{4}} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.13.2

Reescreva ${\sqrt{3}}^{4}$ como $3^{2}$ .

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Etapa 3.1.13.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescrever $\sqrt{3}$ como $3^{\frac{1}{2}}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + 15 \frac{81 {(3^{\frac{1}{2}})}^{4}}{2^{4}} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.13.2.2

Aplique a regra da multiplicação de potências e multiplique os expoentes, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + 15 \frac{81 \cdot 3^{\frac{1}{2} \cdot 4}}{2^{4}} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.13.2.3

Combine $\frac{1}{2}$ e $4$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + 15 \frac{81 \cdot 3^{\frac{4}{2}}}{2^{4}} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.13.2.4

Cancele o fator comum de $4$ e $2$ .

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Etapa 3.1.13.2.4.1

Fatore $2$ de $4$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + 15 \frac{81 \cdot 3^{\frac{2 \cdot 2}{2}}}{2^{4}} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.13.2.4.2

Cancele os fatores comuns.

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Etapa 3.1.13.2.4.2.1

Fatore $2$ de $2$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + 15 \frac{81 \cdot 3^{\frac{2 \cdot 2}{2 (1)}}}{2^{4}} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.13.2.4.2.2

Cancele o fator comum.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + 15 \frac{81 \cdot 3^{\frac{2 \cdot 2}{2 \cdot 1}}}{2^{4}} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.13.2.4.2.3

Reescreva a expressão.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + 15 \frac{81 \cdot 3^{\frac{2}{1}}}{2^{4}} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.13.2.4.2.4

Divida $2$ por $1$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + 15 \frac{81 \cdot 3^{2}}{2^{4}} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.13.3

Eleve $3$ à potência de $2$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + 15 \frac{81 \cdot 9}{2^{4}} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.14

Eleve $2$ à potência de $4$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + 15 \frac{81 \cdot 9}{16} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.15

Multiplique $81$ por $9$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + 15 (\frac{729}{16}) {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.16

Multiplique $15 (\frac{729}{16})$ .

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Etapa 3.1.16.1

Combine $15$ e $\frac{729}{16}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + \frac{15 \cdot 729}{16} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.16.2

Multiplique $15$ por $729$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + \frac{10935}{16} {(\frac{3 i}{2})}^{2} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.17

Use a regra da multiplicação de potências ${(a b)}^{n} = a^{n} b^{n}$ para distribuir o expoente.

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Etapa 3.1.17.1

Aplique a regra do produto a $\frac{3 i}{2}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + \frac{10935}{16} \cdot \frac{{(3 i)}^{2}}{2^{2}} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.17.2

Aplique a regra do produto a $3 i$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + \frac{10935}{16} \cdot \frac{3^{2} i^{2}}{2^{2}} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.18

Combine.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + \frac{10935 (3^{2} i^{2})}{16 \cdot 2^{2}} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.19

Simplifique o numerador.

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Etapa 3.1.19.1

Eleve $3$ à potência de $2$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + \frac{10935 \cdot 9 i^{2}}{16 \cdot 2^{2}} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.19.2

Reescreva $i^{2}$ como $- 1$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + \frac{10935 \cdot 9 \cdot - 1}{16 \cdot 2^{2}} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.19.3

Combine expoentes.

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Etapa 3.1.19.3.1

Multiplique $10935$ por $9$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + \frac{98415 \cdot - 1}{16 \cdot 2^{2}} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.19.3.2

Multiplique $98415$ por $- 1$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + \frac{- 98415}{16 \cdot 2^{2}} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.20

Simplifique o denominador.

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Etapa 3.1.20.1

Reescreva $16$ como $2^{4}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + \frac{- 98415}{2^{4} \cdot 2^{2}} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.20.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + \frac{- 98415}{2^{4 + 2}} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.20.3

Some $4$ e $2$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + \frac{- 98415}{2^{6}} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.21

Eleve $2$ à potência de $6$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} + \frac{- 98415}{64} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.22

Mova o número negativo para a frente da fração.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + 20 {(\frac{3 \sqrt{3}}{2})}^{3} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.23

Use a regra da multiplicação de potências ${(a b)}^{n} = a^{n} b^{n}$ para distribuir o expoente.

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Etapa 3.1.23.1

Aplique a regra do produto a $\frac{3 \sqrt{3}}{2}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + 20 \frac{{(3 \sqrt{3})}^{3}}{2^{3}} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.23.2

Aplique a regra do produto a $3 \sqrt{3}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + 20 \frac{3^{3} {\sqrt{3}}^{3}}{2^{3}} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.24

Simplifique o numerador.

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Etapa 3.1.24.1

Eleve $3$ à potência de $3$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + 20 \frac{27 {\sqrt{3}}^{3}}{2^{3}} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.24.2

Reescreva ${\sqrt{3}}^{3}$ como $\sqrt{3^{3}}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + 20 \frac{27 \sqrt{3^{3}}}{2^{3}} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.24.3

Eleve $3$ à potência de $3$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + 20 \frac{27 \sqrt{27}}{2^{3}} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.24.4

Reescreva $27$ como $3^{2} \cdot 3$ .

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Etapa 3.1.24.4.1

Fatore $9$ de $27$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + 20 \frac{27 \sqrt{9 (3)}}{2^{3}} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.24.4.2

Reescreva $9$ como $3^{2}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + 20 \frac{27 \sqrt{3^{2} \cdot 3}}{2^{3}} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.24.5

Elimine os termos abaixo do radical.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + 20 \frac{27 \cdot 3 \sqrt{3}}{2^{3}} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.24.6

Multiplique $27$ por $3$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + 20 \frac{81 \sqrt{3}}{2^{3}} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.25

Eleve $2$ à potência de $3$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + 20 \frac{81 \sqrt{3}}{8} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.26

Cancele o fator comum de $4$ .

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Etapa 3.1.26.1

Fatore $4$ de $20$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + 4 (5) \frac{81 \sqrt{3}}{8} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.26.2

Fatore $4$ de $8$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + 4 \cdot 5 \frac{81 \sqrt{3}}{4 \cdot 2} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.26.3

Cancele o fator comum.

Etapa 3.1.26.4

Reescreva a expressão.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + 5 \frac{81 \sqrt{3}}{2} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.27

Combine $5$ e $\frac{81 \sqrt{3}}{2}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{5 (81 \sqrt{3})}{2} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.28

Multiplique $81$ por $5$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{405 \sqrt{3}}{2} {(\frac{3 i}{2})}^{3} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.29

Use a regra da multiplicação de potências ${(a b)}^{n} = a^{n} b^{n}$ para distribuir o expoente.

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Etapa 3.1.29.1

Aplique a regra do produto a $\frac{3 i}{2}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{405 \sqrt{3}}{2} \cdot \frac{{(3 i)}^{3}}{2^{3}} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.29.2

Aplique a regra do produto a $3 i$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{405 \sqrt{3}}{2} \cdot \frac{3^{3} i^{3}}{2^{3}} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.30

Combine.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{405 \sqrt{3} (3^{3} i^{3})}{2 \cdot 2^{3}} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.31

Multiplique $2$ por $2^{3}$ somando os expoentes.

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Etapa 3.1.31.1

Multiplique $2$ por $2^{3}$ .

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Etapa 3.1.31.1.1

Eleve $2$ à potência de $1$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{405 \sqrt{3} (3^{3} i^{3})}{2^{1} \cdot 2^{3}} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.31.1.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{405 \sqrt{3} (3^{3} i^{3})}{2^{1 + 3}} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.31.2

Some $1$ e $3$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{405 \sqrt{3} (3^{3} i^{3})}{2^{4}} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.32

Simplifique o numerador.

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Etapa 3.1.32.1

Eleve $3$ à potência de $3$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{405 \sqrt{3} \cdot 27 i^{3}}{2^{4}} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.32.2

Fatore $i^{2}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{405 \sqrt{3} \cdot 27 (i^{2} \cdot i)}{2^{4}} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.32.3

Reescreva $i^{2}$ como $- 1$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{405 \sqrt{3} \cdot 27 (- 1 \cdot i)}{2^{4}} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.32.4

Reescreva $- 1 i$ como $- i$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{405 \sqrt{3} \cdot 27 \cdot - 1 i}{2^{4}} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.32.5

Combine expoentes.

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Etapa 3.1.32.5.1

Fatore o negativo.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{- (405 \sqrt{3} \cdot 27 i)}{2^{4}} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.32.5.2

Multiplique $27$ por $405$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{- (10935 \sqrt{3} i)}{2^{4}} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.32.5.3

Multiplique $10935$ por $- 1$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{- 10935 (\sqrt{3} i)}{2^{4}} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{- 10935 \sqrt{3} i}{2^{4}} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.33

Eleve $2$ à potência de $4$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} + \frac{- 10935 \sqrt{3} i}{16} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.34

Mova o número negativo para a frente da fração.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + 15 {(\frac{3 \sqrt{3}}{2})}^{2} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.35

Use a regra da multiplicação de potências ${(a b)}^{n} = a^{n} b^{n}$ para distribuir o expoente.

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Etapa 3.1.35.1

Aplique a regra do produto a $\frac{3 \sqrt{3}}{2}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + 15 \frac{{(3 \sqrt{3})}^{2}}{2^{2}} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.35.2

Aplique a regra do produto a $3 \sqrt{3}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + 15 \frac{3^{2} {\sqrt{3}}^{2}}{2^{2}} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.36

Simplifique o numerador.

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Etapa 3.1.36.1

Eleve $3$ à potência de $2$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + 15 \frac{9 {\sqrt{3}}^{2}}{2^{2}} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.36.2

Reescreva ${\sqrt{3}}^{2}$ como $3$ .

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Etapa 3.1.36.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescrever $\sqrt{3}$ como $3^{\frac{1}{2}}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + 15 \frac{9 {(3^{\frac{1}{2}})}^{2}}{2^{2}} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.36.2.2

Aplique a regra da multiplicação de potências e multiplique os expoentes, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + 15 \frac{9 \cdot 3^{\frac{1}{2} \cdot 2}}{2^{2}} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.36.2.3

Combine $\frac{1}{2}$ e $2$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + 15 \frac{9 \cdot 3^{\frac{2}{2}}}{2^{2}} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.36.2.4

Cancele o fator comum de $2$ .

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Etapa 3.1.36.2.4.1

Cancele o fator comum.

Etapa 3.1.36.2.4.2

Reescreva a expressão.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + 15 \frac{9 \cdot 3^{1}}{2^{2}} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.36.2.5

Avalie o expoente.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + 15 \frac{9 \cdot 3}{2^{2}} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.37

Eleve $2$ à potência de $2$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + 15 \frac{9 \cdot 3}{4} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.38

Multiplique $9$ por $3$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + 15 (\frac{27}{4}) {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.39

Multiplique $15 (\frac{27}{4})$ .

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Etapa 3.1.39.1

Combine $15$ e $\frac{27}{4}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{15 \cdot 27}{4} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.39.2

Multiplique $15$ por $27$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{405}{4} {(\frac{3 i}{2})}^{4} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.40

Use a regra da multiplicação de potências ${(a b)}^{n} = a^{n} b^{n}$ para distribuir o expoente.

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Etapa 3.1.40.1

Aplique a regra do produto a $\frac{3 i}{2}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{405}{4} \cdot \frac{{(3 i)}^{4}}{2^{4}} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.40.2

Aplique a regra do produto a $3 i$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{405}{4} \cdot \frac{3^{4} i^{4}}{2^{4}} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.41

Combine.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{405 (3^{4} i^{4})}{4 \cdot 2^{4}} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.42

Simplifique o numerador.

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Etapa 3.1.42.1

Eleve $3$ à potência de $4$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{405 \cdot 81 i^{4}}{4 \cdot 2^{4}} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.42.2

Reescreva $i^{4}$ como $1$ .

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Etapa 3.1.42.2.1

Reescreva $i^{4}$ como ${(i^{2})}^{2}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{405 \cdot 81 {(i^{2})}^{2}}{4 \cdot 2^{4}} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.42.2.2

Reescreva $i^{2}$ como $- 1$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{405 \cdot 81 {(- 1)}^{2}}{4 \cdot 2^{4}} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.42.2.3

Eleve $- 1$ à potência de $2$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{405 \cdot 81 \cdot 1}{4 \cdot 2^{4}} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.42.3

Combine expoentes.

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Etapa 3.1.42.3.1

Multiplique $405$ por $81$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805 \cdot 1}{4 \cdot 2^{4}} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.42.3.2

Multiplique $32805$ por $1$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{4 \cdot 2^{4}} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.43

Simplifique o denominador.

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Etapa 3.1.43.1

Reescreva $4$ como $2^{2}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{2^{2} \cdot 2^{4}} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.43.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{2^{2 + 4}} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.43.3

Some $2$ e $4$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{2^{6}} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.44

Eleve $2$ à potência de $6$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + 6 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.45

Cancele o fator comum de $2$ .

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Etapa 3.1.45.1

Fatore $2$ de $6$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + 2 (3) \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.45.2

Cancele o fator comum.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + 2 \cdot 3 \frac{3 \sqrt{3}}{2} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.45.3

Reescreva a expressão.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + 3 (3 \sqrt{3}) {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.46

Multiplique $3$ por $3$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + 9 \sqrt{3} {(\frac{3 i}{2})}^{5} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.47

Use a regra da multiplicação de potências ${(a b)}^{n} = a^{n} b^{n}$ para distribuir o expoente.

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Etapa 3.1.47.1

Aplique a regra do produto a $\frac{3 i}{2}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + 9 \sqrt{3} \frac{{(3 i)}^{5}}{2^{5}} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.47.2

Aplique a regra do produto a $3 i$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + 9 \sqrt{3} \frac{3^{5} i^{5}}{2^{5}} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.48

Simplifique o numerador.

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Etapa 3.1.48.1

Eleve $3$ à potência de $5$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + 9 \sqrt{3} \frac{243 i^{5}}{2^{5}} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.48.2

Fatore $i^{4}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + 9 \sqrt{3} \frac{243 (i^{4} i)}{2^{5}} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.48.3

Reescreva $i^{4}$ como $1$ .

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Etapa 3.1.48.3.1

Reescreva $i^{4}$ como ${(i^{2})}^{2}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + 9 \sqrt{3} \frac{243 ({(i^{2})}^{2} i)}{2^{5}} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.48.3.2

Reescreva $i^{2}$ como $- 1$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + 9 \sqrt{3} \frac{243 ({(- 1)}^{2} i)}{2^{5}} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.48.3.3

Eleve $- 1$ à potência de $2$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + 9 \sqrt{3} \frac{243 (1 i)}{2^{5}} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.48.4

Multiplique $i$ por $1$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + 9 \sqrt{3} \frac{243 i}{2^{5}} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.49

Eleve $2$ à potência de $5$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + 9 \sqrt{3} \frac{243 i}{32} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.50

Multiplique $9 \sqrt{3} \frac{243 i}{32}$ .

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Etapa 3.1.50.1

Combine $\frac{243 i}{32}$ e $9$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + \frac{243 i \cdot 9}{32} \sqrt{3} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.50.2

Multiplique $9$ por $243$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + \frac{2187 i}{32} \sqrt{3} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.50.3

Combine $\frac{2187 i}{32}$ e $\sqrt{3}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + \frac{2187 i \sqrt{3}}{32} + {(\frac{3 i}{2})}^{6}$

Etapa 3.1.51

Use a regra da multiplicação de potências ${(a b)}^{n} = a^{n} b^{n}$ para distribuir o expoente.

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Etapa 3.1.51.1

Aplique a regra do produto a $\frac{3 i}{2}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + \frac{2187 i \sqrt{3}}{32} + \frac{{(3 i)}^{6}}{2^{6}}$

Etapa 3.1.51.2

Aplique a regra do produto a $3 i$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + \frac{2187 i \sqrt{3}}{32} + \frac{3^{6} i^{6}}{2^{6}}$

Etapa 3.1.52

Simplifique o numerador.

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Etapa 3.1.52.1

Eleve $3$ à potência de $6$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + \frac{2187 i \sqrt{3}}{32} + \frac{729 i^{6}}{2^{6}}$

Etapa 3.1.52.2

Fatore $i^{4}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + \frac{2187 i \sqrt{3}}{32} + \frac{729 (i^{4} i^{2})}{2^{6}}$

Etapa 3.1.52.3

Reescreva $i^{4}$ como $1$ .

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Etapa 3.1.52.3.1

Reescreva $i^{4}$ como ${(i^{2})}^{2}$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + \frac{2187 i \sqrt{3}}{32} + \frac{729 ({(i^{2})}^{2} i^{2})}{2^{6}}$

Etapa 3.1.52.3.2

Reescreva $i^{2}$ como $- 1$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + \frac{2187 i \sqrt{3}}{32} + \frac{729 ({(- 1)}^{2} i^{2})}{2^{6}}$

Etapa 3.1.52.3.3

Eleve $- 1$ à potência de $2$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + \frac{2187 i \sqrt{3}}{32} + \frac{729 (1 i^{2})}{2^{6}}$

Etapa 3.1.52.4

Multiplique $i^{2}$ por $1$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + \frac{2187 i \sqrt{3}}{32} + \frac{729 i^{2}}{2^{6}}$

Etapa 3.1.52.5

Reescreva $i^{2}$ como $- 1$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + \frac{2187 i \sqrt{3}}{32} + \frac{729 \cdot - 1}{2^{6}}$

Etapa 3.1.53

Eleve $2$ à potência de $6$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + \frac{2187 i \sqrt{3}}{32} + \frac{729 \cdot - 1}{64}$

Etapa 3.1.54

Multiplique $729$ por $- 1$ .

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + \frac{2187 i \sqrt{3}}{32} + \frac{- 729}{64}$

Etapa 3.1.55

Mova o número negativo para a frente da fração.

$\frac{19683}{64} + \frac{19683 \sqrt{3} i}{32} - \frac{98415}{64} - \frac{10935 \sqrt{3} i}{16} + \frac{32805}{64} + \frac{2187 i \sqrt{3}}{32} - \frac{729}{64}$

Etapa 3.2

Simplifique os termos.

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Etapa 3.2.1

Combine os numeradores em relação ao denominador comum.

$\frac{19683 - 98415 + 32805 - 729}{64} + \frac{19683 \sqrt{3} i + 2187 i \sqrt{3}}{32} + \frac{- 10935 \sqrt{3} i}{16}$

Etapa 3.2.2

Simplifique a expressão.

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Etapa 3.2.2.1

Subtraia $98415$ de $19683$ .

$\frac{- 78732 + 32805 - 729}{64} + \frac{19683 \sqrt{3} i + 2187 i \sqrt{3}}{32} + \frac{- 10935 \sqrt{3} i}{16}$

Etapa 3.2.2.2

Some $- 78732$ e $32805$ .

$\frac{- 45927 - 729}{64} + \frac{19683 \sqrt{3} i + 2187 i \sqrt{3}}{32} + \frac{- 10935 \sqrt{3} i}{16}$

Etapa 3.2.2.3

Subtraia $729$ de $- 45927$ .

$\frac{- 46656}{64} + \frac{19683 \sqrt{3} i + 2187 i \sqrt{3}}{32} + \frac{- 10935 \sqrt{3} i}{16}$

Etapa 3.2.2.4

Reordene os fatores de $19683 \sqrt{3} i$ .

$\frac{- 46656}{64} + \frac{19683 i \sqrt{3} + 2187 i \sqrt{3}}{32} + \frac{- 10935 \sqrt{3} i}{16}$

Etapa 3.2.3

Some $19683 i \sqrt{3}$ e $2187 i \sqrt{3}$ .

$\frac{- 46656}{64} + \frac{21870 i \sqrt{3}}{32} + \frac{- 10935 \sqrt{3} i}{16}$

Etapa 3.3

Simplifique cada termo.

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Etapa 3.3.1

Divida $- 46656$ por $64$ .

$- 729 + \frac{21870 i \sqrt{3}}{32} + \frac{- 10935 \sqrt{3} i}{16}$

Etapa 3.3.2

Cancele o fator comum de $21870$ e $32$ .

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Etapa 3.3.2.1

Fatore $2$ de $21870 i \sqrt{3}$ .

$- 729 + \frac{2 (10935 i \sqrt{3})}{32} + \frac{- 10935 \sqrt{3} i}{16}$

Etapa 3.3.2.2

Cancele os fatores comuns.

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Etapa 3.3.2.2.1

Fatore $2$ de $32$ .

$- 729 + \frac{2 (10935 i \sqrt{3})}{2 (16)} + \frac{- 10935 \sqrt{3} i}{16}$

Etapa 3.3.2.2.2

Cancele o fator comum.

$- 729 + \frac{2 (10935 i \sqrt{3})}{2 \cdot 16} + \frac{- 10935 \sqrt{3} i}{16}$

Etapa 3.3.2.2.3

Reescreva a expressão.

$- 729 + \frac{10935 i \sqrt{3}}{16} + \frac{- 10935 \sqrt{3} i}{16}$

Etapa 3.3.3

Mova o número negativo para a frente da fração.

$- 729 + \frac{10935 i \sqrt{3}}{16} - \frac{10935 \sqrt{3} i}{16}$

Etapa 3.4

Simplifique os termos.

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Etapa 3.4.1

Combine os numeradores em relação ao denominador comum.

$- 729 + \frac{10935 i \sqrt{3} - 10935 \sqrt{3} i}{16}$

Etapa 3.4.2

Reordene os fatores de $- 10935 \sqrt{3} i$ .

$- 729 + \frac{10935 i \sqrt{3} - 10935 i \sqrt{3}}{16}$

Etapa 3.4.3

Subtraia $10935 i \sqrt{3}$ de $10935 i \sqrt{3}$ .

$- 729 + \frac{0}{16}$

Etapa 3.4.4

Simplifique a expressão.

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Etapa 3.4.4.1

Divida $0$ por $16$ .

$- 729 + 0$

Etapa 3.4.4.2

Some $- 729$ e $0$ .

$- 729$