Cálculo Ejemplos

${(\cos (\frac{3 π}{12}) + i \sin (\frac{3 π}{12}))}^{5}$

Paso 1

Simplifica cada término.

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Paso 1.1

Cancela el factor común de $3$ y $12$ .

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Paso 1.1.1

Factoriza $3$ de $3 π$ .

${(\cos (\frac{3 (π)}{12}) + i \sin (\frac{3 π}{12}))}^{5}$

Paso 1.1.2

Cancela los factores comunes.

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Paso 1.1.2.1

Factoriza $3$ de $12$ .

${(\cos (\frac{3 π}{3 \cdot 4}) + i \sin (\frac{3 π}{12}))}^{5}$

Paso 1.1.2.2

Cancela el factor común.

${(\cos (\frac{3 π}{3 \cdot 4}) + i \sin (\frac{3 π}{12}))}^{5}$

Paso 1.1.2.3

Reescribe la expresión.

${(\cos (\frac{π}{4}) + i \sin (\frac{3 π}{12}))}^{5}$

Paso 1.2

El valor exacto de $\cos (\frac{π}{4})$ es $\frac{\sqrt{2}}{2}$ .

${(\frac{\sqrt{2}}{2} + i \sin (\frac{3 π}{12}))}^{5}$

Paso 1.3

Cancela el factor común de $3$ y $12$ .

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Paso 1.3.1

Factoriza $3$ de $3 π$ .

${(\frac{\sqrt{2}}{2} + i \sin (\frac{3 (π)}{12}))}^{5}$

Paso 1.3.2

Cancela los factores comunes.

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Paso 1.3.2.1

Factoriza $3$ de $12$ .

${(\frac{\sqrt{2}}{2} + i \sin (\frac{3 π}{3 \cdot 4}))}^{5}$

Paso 1.3.2.2

Cancela el factor común.

${(\frac{\sqrt{2}}{2} + i \sin (\frac{3 π}{3 \cdot 4}))}^{5}$

Paso 1.3.2.3

Reescribe la expresión.

${(\frac{\sqrt{2}}{2} + i \sin (\frac{π}{4}))}^{5}$

Paso 1.4

El valor exacto de $\sin (\frac{π}{4})$ es $\frac{\sqrt{2}}{2}$ .

${(\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2})}^{5}$

Paso 1.5

Combina $i$ y $\frac{\sqrt{2}}{2}$ .

${(\frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2})}^{5}$

Paso 2

Usa el teorema del binomio.

${(\frac{\sqrt{2}}{2})}^{5} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3

Simplifica los términos.

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Paso 3.1

Simplifica cada término.

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Paso 3.1.1

Aplica la regla del producto a $\frac{\sqrt{2}}{2}$ .

$\frac{{\sqrt{2}}^{5}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.2

Simplifica el numerador.

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Paso 3.1.2.1

Reescribe ${\sqrt{2}}^{5}$ como $\sqrt{2^{5}}$ .

$\frac{\sqrt{2^{5}}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.2.2

Eleva $2$ a la potencia de $5$ .

$\frac{\sqrt{32}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.2.3

Reescribe $32$ como $4^{2} \cdot 2$ .

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Paso 3.1.2.3.1

Factoriza $16$ de $32$ .

$\frac{\sqrt{16 (2)}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.2.3.2

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

$\frac{\sqrt{4^{2} \cdot 2}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.2.4

Retira los términos de abajo del radical.

$\frac{4 \sqrt{2}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.3

Eleva $2$ a la potencia de $5$ .

$\frac{4 \sqrt{2}}{32} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.4

Cancela el factor común de $4$ y $32$ .

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Paso 3.1.4.1

Factoriza $4$ de $4 \sqrt{2}$ .

$\frac{4 (\sqrt{2})}{32} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.4.2

Cancela los factores comunes.

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Paso 3.1.4.2.1

Factoriza $4$ de $32$ .

$\frac{4 \sqrt{2}}{4 \cdot 8} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.4.2.2

Cancela el factor común.

Paso 3.1.4.2.3

Reescribe la expresión.

$\frac{\sqrt{2}}{8} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.5

Aplica la regla del producto a $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{8} + 5 \frac{{\sqrt{2}}^{4}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.6

Simplifica el numerador.

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Paso 3.1.6.1

Reescribe ${\sqrt{2}}^{4}$ como $2^{2}$ .

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Paso 3.1.6.1.1

Usa $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescribir $\sqrt{2}$ como $2^{\frac{1}{2}}$ .

$\frac{\sqrt{2}}{8} + 5 \frac{{(2^{\frac{1}{2}})}^{4}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.6.1.2

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{1}{2} \cdot 4}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.6.1.3

Combina $\frac{1}{2}$ y $4$ .

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{4}{2}}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.6.1.4

Cancela el factor común de $4$ y $2$ .

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Paso 3.1.6.1.4.1

Factoriza $2$ de $4$ .

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{2 \cdot 2}{2}}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.6.1.4.2

Cancela los factores comunes.

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Paso 3.1.6.1.4.2.1

Factoriza $2$ de $2$ .

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{2 \cdot 2}{2 (1)}}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.6.1.4.2.2

Cancela el factor común.

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{2 \cdot 2}{2 \cdot 1}}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.6.1.4.2.3

Reescribe la expresión.

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{2}{1}}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.6.1.4.2.4

Divide $2$ por $1$ .

$\frac{\sqrt{2}}{8} + 5 \frac{2^{2}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.6.2

Eleva $2$ a la potencia de $2$ .

$\frac{\sqrt{2}}{8} + 5 \frac{4}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.7

Eleva $2$ a la potencia de $4$ .

$\frac{\sqrt{2}}{8} + 5 (\frac{4}{16}) \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.8

Cancela el factor común de $4$ y $16$ .

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Paso 3.1.8.1

Factoriza $4$ de $4$ .

$\frac{\sqrt{2}}{8} + 5 \frac{4 (1)}{16} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.8.2

Cancela los factores comunes.

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Paso 3.1.8.2.1

Factoriza $4$ de $16$ .

$\frac{\sqrt{2}}{8} + 5 \frac{4 \cdot 1}{4 \cdot 4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.8.2.2

Cancela el factor común.

Paso 3.1.8.2.3

Reescribe la expresión.

$\frac{\sqrt{2}}{8} + 5 (\frac{1}{4}) \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.9

Combina $5$ y $\frac{1}{4}$ .

$\frac{\sqrt{2}}{8} + \frac{5}{4} \cdot \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.10

Multiplica $\frac{5}{4} \cdot \frac{i \sqrt{2}}{2}$ .

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Paso 3.1.10.1

Multiplica $\frac{5}{4}$ por $\frac{i \sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 (i \sqrt{2})}{4 \cdot 2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.10.2

Multiplica $4$ por $2$ .

$\frac{\sqrt{2}}{8} + \frac{5 (i \sqrt{2})}{8} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.11

Aplica la regla del producto a $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{{\sqrt{2}}^{3}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.12

Simplifica el numerador.

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Paso 3.1.12.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{\sqrt{2^{3}}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.12.2

Eleva $2$ a la potencia de $3$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{\sqrt{8}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.12.3

Reescribe $8$ como $2^{2} \cdot 2$ .

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Paso 3.1.12.3.1

Factoriza $4$ de $8$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{\sqrt{4 (2)}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.12.3.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{\sqrt{2^{2} \cdot 2}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.12.4

Retira los términos de abajo del radical.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{2 \sqrt{2}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.13

Eleva $2$ a la potencia de $3$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{2 \sqrt{2}}{8} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.14

Cancela el factor común de $2$ .

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Paso 3.1.14.1

Factoriza $2$ de $10$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 2 (5) \frac{2 \sqrt{2}}{8} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.14.2

Factoriza $2$ de $8$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 2 \cdot 5 \frac{2 \sqrt{2}}{2 \cdot 4} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.14.3

Cancela el factor común.

Paso 3.1.14.4

Reescribe la expresión.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 5 \frac{2 \sqrt{2}}{4} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.15

Combina $5$ y $\frac{2 \sqrt{2}}{4}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 (2 \sqrt{2})}{4} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.16

Multiplica $2$ por $5$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{10 \sqrt{2}}{4} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.17

Cancela el factor común de $10$ y $4$ .

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Paso 3.1.17.1

Factoriza $2$ de $10 \sqrt{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (5 \sqrt{2})}{4} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.17.2

Cancela los factores comunes.

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Paso 3.1.17.2.1

Factoriza $2$ de $4$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (5 \sqrt{2})}{2 (2)} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.17.2.2

Cancela el factor común.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (5 \sqrt{2})}{2 \cdot 2} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.17.2.3

Reescribe la expresión.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.18

Usa la regla de la potencia ${(a b)}^{n} = a^{n} b^{n}$ para distribuir el exponente.

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Paso 3.1.18.1

Aplica la regla del producto a $\frac{i \sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2}}{2} \cdot \frac{{(i \sqrt{2})}^{2}}{2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.18.2

Aplica la regla del producto a $i \sqrt{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2}}{2} \cdot \frac{i^{2} {\sqrt{2}}^{2}}{2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.19

Combinar.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2} (i^{2} {\sqrt{2}}^{2})}{2 \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.20

Multiplica $\sqrt{2}$ por ${\sqrt{2}}^{2}$ sumando los exponentes.

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Paso 3.1.20.1

Mueve ${\sqrt{2}}^{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 ({\sqrt{2}}^{2} \sqrt{2}) i^{2}}{2 \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.20.2

Multiplica ${\sqrt{2}}^{2}$ por $\sqrt{2}$ .

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Paso 3.1.20.2.1

Eleva $\sqrt{2}$ a la potencia de $1$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 ({\sqrt{2}}^{2} {\sqrt{2}}^{1}) i^{2}}{2 \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.20.2.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 {\sqrt{2}}^{2 + 1} i^{2}}{2 \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.20.3

Suma $2$ y $1$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 {\sqrt{2}}^{3} i^{2}}{2 \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.21

Multiplica $2$ por $2^{2}$ sumando los exponentes.

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Paso 3.1.21.1

Multiplica $2$ por $2^{2}$ .

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Paso 3.1.21.1.1

Eleva $2$ a la potencia de $1$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 {\sqrt{2}}^{3} i^{2}}{2^{1} \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.21.1.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 {\sqrt{2}}^{3} i^{2}}{2^{1 + 2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.21.2

Suma $1$ y $2$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 {\sqrt{2}}^{3} i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.22

Simplifica el numerador.

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Paso 3.1.22.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2^{3}} i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.22.2

Eleva $2$ a la potencia de $3$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{8} i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.22.3

Reescribe $8$ como $2^{2} \cdot 2$ .

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Paso 3.1.22.3.1

Factoriza $4$ de $8$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{4 (2)} i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.22.3.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2^{2} \cdot 2} i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.22.4

Retira los términos de abajo del radical.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 (2 \sqrt{2}) i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.22.5

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \cdot 2 \sqrt{2} \cdot - 1}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.22.6

Combina exponentes.

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Paso 3.1.22.6.1

Multiplica $5$ por $2$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{10 \sqrt{2} \cdot - 1}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.22.6.2

Multiplica $- 1$ por $10$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{- 10 \sqrt{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.23

Eleva $2$ a la potencia de $3$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{- 10 \sqrt{2}}{8} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.24

Cancela el factor común de $- 10$ y $8$ .

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Paso 3.1.24.1

Factoriza $2$ de $- 10 \sqrt{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (- 5 \sqrt{2})}{8} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.24.2

Cancela los factores comunes.

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Paso 3.1.24.2.1

Factoriza $2$ de $8$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (- 5 \sqrt{2})}{2 (4)} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.24.2.2

Cancela el factor común.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (- 5 \sqrt{2})}{2 \cdot 4} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.24.2.3

Reescribe la expresión.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{4} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.25

Mueve el negativo al frente de la fracción.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{(5) \sqrt{2}}{4} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.26

Aplica la regla del producto a $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{{\sqrt{2}}^{2}}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.27

Reescribe ${\sqrt{2}}^{2}$ como $2$ .

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Paso 3.1.27.1

Usa $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescribir $\sqrt{2}$ como $2^{\frac{1}{2}}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.27.2

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.27.3

Combina $\frac{1}{2}$ y $2$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{2^{\frac{2}{2}}}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.27.4

Cancela el factor común de $2$ .

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Paso 3.1.27.4.1

Cancela el factor común.

Paso 3.1.27.4.2

Reescribe la expresión.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{2^{1}}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.27.5

Evalúa el exponente.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{2}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.28

Eleva $2$ a la potencia de $2$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 (\frac{2}{4}) {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.29

Cancela el factor común de $2$ .

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Paso 3.1.29.1

Factoriza $2$ de $10$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 2 (5) \frac{2}{4} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.29.2

Factoriza $2$ de $4$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 2 \cdot 5 \frac{2}{2 \cdot 2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.29.3

Cancela el factor común.

Paso 3.1.29.4

Reescribe la expresión.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 (\frac{2}{2}) {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.30

Combina $5$ y $\frac{2}{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + \frac{5 \cdot 2}{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.31

Multiplica $5$ por $2$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + \frac{10}{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.32

Divide $10$ por $2$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.33

Usa la regla de la potencia ${(a b)}^{n} = a^{n} b^{n}$ para distribuir el exponente.

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Paso 3.1.33.1

Aplica la regla del producto a $\frac{i \sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{{(i \sqrt{2})}^{3}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.33.2

Aplica la regla del producto a $i \sqrt{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{i^{3} {\sqrt{2}}^{3}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.34

Simplifica el numerador.

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Paso 3.1.34.1

Factoriza $i^{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{i^{2} \cdot i {\sqrt{2}}^{3}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.34.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- 1 \cdot i {\sqrt{2}}^{3}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.34.3

Reescribe $- 1 i$ como $- i$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i {\sqrt{2}}^{3}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.34.4

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \sqrt{2^{3}}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.34.5

Eleva $2$ a la potencia de $3$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \sqrt{8}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.34.6

Reescribe $8$ como $2^{2} \cdot 2$ .

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Paso 3.1.34.6.1

Factoriza $4$ de $8$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \sqrt{4 (2)}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.34.6.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \sqrt{2^{2} \cdot 2}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.34.7

Retira los términos de abajo del radical.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \cdot 2 \sqrt{2}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.34.8

Multiplica $2$ por $- 1$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- 2 i \sqrt{2}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.35

Eleva $2$ a la potencia de $3$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- 2 i \sqrt{2}}{8} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.36

Cancela el factor común de $- 2$ y $8$ .

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Paso 3.1.36.1

Factoriza $2$ de $- 2 i \sqrt{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{2 (- i \sqrt{2})}{8} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.36.2

Cancela los factores comunes.

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Paso 3.1.36.2.1

Factoriza $2$ de $8$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{2 (- i \sqrt{2})}{2 (4)} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.36.2.2

Cancela el factor común.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{2 (- i \sqrt{2})}{2 \cdot 4} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.36.2.3

Reescribe la expresión.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \sqrt{2}}{4} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.37

Mueve el negativo al frente de la fracción.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 (- \frac{i \sqrt{2}}{4}) + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.38

Multiplica $5 (- \frac{i \sqrt{2}}{4})$ .

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Paso 3.1.38.1

Multiplica $- 1$ por $5$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - 5 \frac{i \sqrt{2}}{4} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.38.2

Combina $- 5$ y $\frac{i \sqrt{2}}{4}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + \frac{- 5 (i \sqrt{2})}{4} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.39

Mueve el negativo al frente de la fracción.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{(5) i \sqrt{2}}{4} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.40

Combina $5$ y $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.41

Usa la regla de la potencia ${(a b)}^{n} = a^{n} b^{n}$ para distribuir el exponente.

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Paso 3.1.41.1

Aplica la regla del producto a $\frac{i \sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{2} \cdot \frac{{(i \sqrt{2})}^{4}}{2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.41.2

Aplica la regla del producto a $i \sqrt{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{2} \cdot \frac{i^{4} {\sqrt{2}}^{4}}{2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.42

Combinar.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2} (i^{4} {\sqrt{2}}^{4})}{2 \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.43

Multiplica $\sqrt{2}$ por ${\sqrt{2}}^{4}$ sumando los exponentes.

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Paso 3.1.43.1

Mueve ${\sqrt{2}}^{4}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 ({\sqrt{2}}^{4} \sqrt{2}) i^{4}}{2 \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.43.2

Multiplica ${\sqrt{2}}^{4}$ por $\sqrt{2}$ .

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Paso 3.1.43.2.1

Eleva $\sqrt{2}$ a la potencia de $1$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 ({\sqrt{2}}^{4} {\sqrt{2}}^{1}) i^{4}}{2 \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.43.2.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 {\sqrt{2}}^{4 + 1} i^{4}}{2 \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.43.3

Suma $4$ y $1$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 {\sqrt{2}}^{5} i^{4}}{2 \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.44

Multiplica $2$ por $2^{4}$ sumando los exponentes.

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Paso 3.1.44.1

Multiplica $2$ por $2^{4}$ .

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Paso 3.1.44.1.1

Eleva $2$ a la potencia de $1$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 {\sqrt{2}}^{5} i^{4}}{2^{1} \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.44.1.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 {\sqrt{2}}^{5} i^{4}}{2^{1 + 4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.44.2

Suma $1$ y $4$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 {\sqrt{2}}^{5} i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.45

Simplifica el numerador.

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Paso 3.1.45.1

Reescribe ${\sqrt{2}}^{5}$ como $\sqrt{2^{5}}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2^{5}} i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.45.2

Eleva $2$ a la potencia de $5$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{32} i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.45.3

Reescribe $32$ como $4^{2} \cdot 2$ .

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Paso 3.1.45.3.1

Factoriza $16$ de $32$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{16 (2)} i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.45.3.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{4^{2} \cdot 2} i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.45.4

Retira los términos de abajo del radical.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 (4 \sqrt{2}) i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.45.5

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

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Paso 3.1.45.5.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 (4 \sqrt{2}) {(i^{2})}^{2}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.45.5.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 (4 \sqrt{2}) {(- 1)}^{2}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.45.5.3

Eleva $- 1$ a la potencia de $2$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 (4 \sqrt{2}) \cdot 1}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \cdot 4 \sqrt{2} \cdot 1}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.45.6

Combina exponentes.

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Paso 3.1.45.6.1

Multiplica $5$ por $4$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{20 \sqrt{2} \cdot 1}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.45.6.2

Multiplica $20$ por $1$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{20 \sqrt{2}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.46

Eleva $2$ a la potencia de $5$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{20 \sqrt{2}}{32} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.47

Cancela el factor común de $20$ y $32$ .

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Paso 3.1.47.1

Factoriza $4$ de $20 \sqrt{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{4 (5 \sqrt{2})}{32} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.47.2

Cancela los factores comunes.

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Paso 3.1.47.2.1

Factoriza $4$ de $32$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{4 (5 \sqrt{2})}{4 (8)} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.47.2.2

Cancela el factor común.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{4 (5 \sqrt{2})}{4 \cdot 8} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.47.2.3

Reescribe la expresión.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + {(\frac{i \sqrt{2}}{2})}^{5}$

Paso 3.1.48

Usa la regla de la potencia ${(a b)}^{n} = a^{n} b^{n}$ para distribuir el exponente.

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Paso 3.1.48.1

Aplica la regla del producto a $\frac{i \sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{{(i \sqrt{2})}^{5}}{2^{5}}$

Paso 3.1.48.2

Aplica la regla del producto a $i \sqrt{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i^{5} {\sqrt{2}}^{5}}{2^{5}}$

Paso 3.1.49

Simplifica el numerador.

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Paso 3.1.49.1

Factoriza $i^{4}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i^{4} i {\sqrt{2}}^{5}}{2^{5}}$

Paso 3.1.49.2

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

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Paso 3.1.49.2.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{{(i^{2})}^{2} i {\sqrt{2}}^{5}}{2^{5}}$

Paso 3.1.49.2.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{{(- 1)}^{2} i {\sqrt{2}}^{5}}{2^{5}}$

Paso 3.1.49.2.3

Eleva $- 1$ a la potencia de $2$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{1 i {\sqrt{2}}^{5}}{2^{5}}$

Paso 3.1.49.3

Multiplica $i$ por $1$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i {\sqrt{2}}^{5}}{2^{5}}$

Paso 3.1.49.4

Reescribe ${\sqrt{2}}^{5}$ como $\sqrt{2^{5}}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \sqrt{2^{5}}}{2^{5}}$

Paso 3.1.49.5

Eleva $2$ a la potencia de $5$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \sqrt{32}}{2^{5}}$

Paso 3.1.49.6

Reescribe $32$ como $4^{2} \cdot 2$ .

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Paso 3.1.49.6.1

Factoriza $16$ de $32$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \sqrt{16 (2)}}{2^{5}}$

Paso 3.1.49.6.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \sqrt{4^{2} \cdot 2}}{2^{5}}$

Paso 3.1.49.7

Retira los términos de abajo del radical.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \cdot 4 \sqrt{2}}{2^{5}}$

Paso 3.1.50

Eleva $2$ a la potencia de $5$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \cdot 4 \sqrt{2}}{32}$

Paso 3.1.51

Mueve $4$ a la izquierda de $i$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{4 i \sqrt{2}}{32}$

Paso 3.1.52

Cancela el factor común de $4$ y $32$ .

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Paso 3.1.52.1

Factoriza $4$ de $4 i \sqrt{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{4 (i \sqrt{2})}{32}$

Paso 3.1.52.2

Cancela los factores comunes.

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Paso 3.1.52.2.1

Factoriza $4$ de $32$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{4 (i \sqrt{2})}{4 \cdot 8}$

Paso 3.1.52.2.2

Cancela el factor común.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{4 (i \sqrt{2})}{4 \cdot 8}$

Paso 3.1.52.2.3

Reescribe la expresión.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \sqrt{2}}{8}$

Paso 3.2

Simplifica los términos.

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Paso 3.2.1

Combina los numeradores sobre el denominador común.

$\frac{\sqrt{2} + 5 i \sqrt{2} + 5 \sqrt{2} + i \sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{4} + \frac{- 5 i \sqrt{2}}{4}$

Paso 3.2.2

Suma $\sqrt{2}$ y $5 \sqrt{2}$ .

$\frac{5 i \sqrt{2} + 6 \sqrt{2} + i \sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{4} + \frac{- 5 i \sqrt{2}}{4}$

Paso 3.2.3

Suma $5 i \sqrt{2}$ y $i \sqrt{2}$ .

$\frac{6 i \sqrt{2} + 6 \sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{4} + \frac{- 5 i \sqrt{2}}{4}$

Paso 3.2.4

Reordena $6 i \sqrt{2}$ y $6 \sqrt{2}$ .

$\frac{6 \sqrt{2} + 6 i \sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{4} + \frac{- 5 i \sqrt{2}}{4}$

Paso 3.3

Simplifica cada término.

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Paso 3.3.1

Mueve el negativo al frente de la fracción.

$\frac{6 \sqrt{2} + 6 i \sqrt{2}}{8} - \frac{(5) \sqrt{2}}{4} + \frac{- 5 i \sqrt{2}}{4}$

Paso 3.3.2

Mueve el negativo al frente de la fracción.

$\frac{6 \sqrt{2} + 6 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4}$

Paso 3.4

Reordena $\frac{6 \sqrt{2} + 6 i \sqrt{2}}{8}$ y $- \frac{5 \sqrt{2}}{4}$ .

$- \frac{5 \sqrt{2}}{4} + \frac{6 \sqrt{2} + 6 i \sqrt{2}}{8} - \frac{5 i \sqrt{2}}{4}$

Paso 4

Esta es la forma trigonométrica de un número complejo donde $| z |$ es el módulo y $θ$ es el ángulo creado en el plano complejo.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Paso 5

El módulo de un número complejo es la distancia desde el origen en el plano complejo.

$| z | = \sqrt{a^{2} + b^{2}}$ donde $z = a + b i$

Paso 6

Sustituye los valores reales de $a = - \frac{5 \sqrt{2}}{4}$ y $b = \frac{1}{8}$ .

$| z | = \sqrt{{(\frac{1}{8})}^{2} + {(- \frac{5 \sqrt{2}}{4})}^{2}}$

Paso 7

Obtén $| z |$ .

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Paso 7.1

Aplica la regla del producto a $\frac{1}{8}$ .

$| z | = \sqrt{\frac{1^{2}}{8^{2}} + {(- \frac{5 \sqrt{2}}{4})}^{2}}$

Paso 7.2

Uno elevado a cualquier potencia es uno.

$| z | = \sqrt{\frac{1}{8^{2}} + {(- \frac{5 \sqrt{2}}{4})}^{2}}$

Paso 7.3

Eleva $8$ a la potencia de $2$ .

$| z | = \sqrt{\frac{1}{64} + {(- \frac{5 \sqrt{2}}{4})}^{2}}$

Paso 7.4

Usa la regla de la potencia ${(a b)}^{n} = a^{n} b^{n}$ para distribuir el exponente.

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Paso 7.4.1

Aplica la regla del producto a $- \frac{5 \sqrt{2}}{4}$ .

$| z | = \sqrt{\frac{1}{64} + {(- 1)}^{2} {(\frac{5 \sqrt{2}}{4})}^{2}}$

Paso 7.4.2

Aplica la regla del producto a $\frac{5 \sqrt{2}}{4}$ .

$| z | = \sqrt{\frac{1}{64} + {(- 1)}^{2} (\frac{{(5 \sqrt{2})}^{2}}{4^{2}})}$

Paso 7.4.3

Aplica la regla del producto a $5 \sqrt{2}$ .

$| z | = \sqrt{\frac{1}{64} + {(- 1)}^{2} (\frac{5^{2} {\sqrt{2}}^{2}}{4^{2}})}$

Paso 7.5

Simplifica la expresión.

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Paso 7.5.1

Eleva $- 1$ a la potencia de $2$ .

$| z | = \sqrt{\frac{1}{64} + 1 (\frac{5^{2} {\sqrt{2}}^{2}}{4^{2}})}$

Paso 7.5.2

Multiplica $\frac{5^{2} {\sqrt{2}}^{2}}{4^{2}}$ por $1$ .

$| z | = \sqrt{\frac{1}{64} + \frac{5^{2} {\sqrt{2}}^{2}}{4^{2}}}$

Paso 7.6

Simplifica el numerador.

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Paso 7.6.1

Eleva $5$ a la potencia de $2$ .

$| z | = \sqrt{\frac{1}{64} + \frac{25 {\sqrt{2}}^{2}}{4^{2}}}$

Paso 7.6.2

Reescribe ${\sqrt{2}}^{2}$ como $2$ .

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Paso 7.6.2.1

Usa $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescribir $\sqrt{2}$ como $2^{\frac{1}{2}}$ .

$| z | = \sqrt{\frac{1}{64} + \frac{25 {(2^{\frac{1}{2}})}^{2}}{4^{2}}}$

Paso 7.6.2.2

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2^{\frac{1}{2} \cdot 2}}{4^{2}}}$

Paso 7.6.2.3

Combina $\frac{1}{2}$ y $2$ .

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2^{\frac{2}{2}}}{4^{2}}}$

Paso 7.6.2.4

Cancela el factor común de $2$ .

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Paso 7.6.2.4.1

Cancela el factor común.

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2^{\frac{2}{2}}}{4^{2}}}$

Paso 7.6.2.4.2

Reescribe la expresión.

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2}{4^{2}}}$

Paso 7.6.2.5

Evalúa el exponente.

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2}{4^{2}}}$

Paso 7.7

Reduce la expresión mediante la cancelación de los factores comunes.

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Paso 7.7.1

Eleva $4$ a la potencia de $2$ .

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2}{16}}$

Paso 7.7.2

Multiplica $25$ por $2$ .

$| z | = \sqrt{\frac{1}{64} + \frac{50}{16}}$

Paso 7.7.3

Cancela el factor común de $50$ y $16$ .

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Paso 7.7.3.1

Factoriza $2$ de $50$ .

$| z | = \sqrt{\frac{1}{64} + \frac{2 (25)}{16}}$

Paso 7.7.3.2

Cancela los factores comunes.

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Paso 7.7.3.2.1

Factoriza $2$ de $16$ .

$| z | = \sqrt{\frac{1}{64} + \frac{2 \cdot 25}{2 \cdot 8}}$

Paso 7.7.3.2.2

Cancela el factor común.

$| z | = \sqrt{\frac{1}{64} + \frac{2 \cdot 25}{2 \cdot 8}}$

Paso 7.7.3.2.3

Reescribe la expresión.

$| z | = \sqrt{\frac{1}{64} + \frac{25}{8}}$

Paso 7.8

Para escribir $\frac{25}{8}$ como una fracción con un denominador común, multiplica por $\frac{8}{8}$ .

$| z | = \sqrt{\frac{1}{64} + \frac{25}{8} \cdot \frac{8}{8}}$

Paso 7.9

Escribe cada expresión con un denominador común de $64$ , mediante la multiplicación de cada uno por un factor adecuado de $1$ .

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Paso 7.9.1

Multiplica $\frac{25}{8}$ por $\frac{8}{8}$ .

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 8}{8 \cdot 8}}$

Paso 7.9.2

Multiplica $8$ por $8$ .

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 8}{64}}$

Paso 7.10

Combina los numeradores sobre el denominador común.

$| z | = \sqrt{\frac{1 + 25 \cdot 8}{64}}$

Paso 7.11

Simplifica el numerador.

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Paso 7.11.1

Multiplica $25$ por $8$ .

$| z | = \sqrt{\frac{1 + 200}{64}}$

Paso 7.11.2

Suma $1$ y $200$ .

$| z | = \sqrt{\frac{201}{64}}$

Paso 7.12

Reescribe $\sqrt{\frac{201}{64}}$ como $\frac{\sqrt{201}}{\sqrt{64}}$ .

$| z | = \frac{\sqrt{201}}{\sqrt{64}}$

Paso 7.13

Simplifica el denominador.

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Paso 7.13.1

Reescribe $64$ como $8^{2}$ .

$| z | = \frac{\sqrt{201}}{\sqrt{8^{2}}}$

Paso 7.13.2

Extrae los términos de abajo del radical, bajo el supuesto de que tienes números reales positivos.

$| z | = \frac{\sqrt{201}}{8}$

Paso 8

El ángulo del punto en el plano complejo es la inversa de la tangente de la parte compleja en la parte real.

$θ = \arctan (\frac{\frac{1}{8}}{- \frac{5 \sqrt{2}}{4}})$

Paso 9

Como la tangente inversa de $\frac{\frac{1}{8}}{- \frac{5 \sqrt{2}}{4}}$ produce un ángulo en el segundo cuadrante, el valor del ángulo es $3.07099947$ .

$θ = 3.07099947$

Paso 10

Sustituye los valores de $θ = 3.07099947$ y $| z | = \frac{\sqrt{201}}{8}$ .

$\frac{\sqrt{201}}{8 (\cos (3.07099947) + i \sin (3.07099947))}$