Pré-cálculo Exemplos

$\sin^{4} (3 x)$

Etapa 1

Aplique a fórmula do arco triplo do seno.

${(- 4 \sin^{3} (x) + 3 \sin (x))}^{4}$

Etapa 2

Use o teorema binomial.

${(- 4 \sin^{3} (x))}^{4} + 4 {(- 4 \sin^{3} (x))}^{3} (3 \sin (x)) + 6 {(- 4 \sin^{3} (x))}^{2} {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3

Simplifique cada termo.

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Etapa 3.1

Aplique a regra do produto a $- 4 \sin^{3} (x)$ .

${(- 4)}^{4} {(\sin^{3} (x))}^{4} + 4 {(- 4 \sin^{3} (x))}^{3} (3 \sin (x)) + 6 {(- 4 \sin^{3} (x))}^{2} {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.2

Eleve $- 4$ à potência de $4$ .

$256 {(\sin^{3} (x))}^{4} + 4 {(- 4 \sin^{3} (x))}^{3} (3 \sin (x)) + 6 {(- 4 \sin^{3} (x))}^{2} {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.3

Multiplique os expoentes em ${(\sin^{3} (x))}^{4}$ .

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Etapa 3.3.1

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

$256 {\sin (x)}^{3 \cdot 4} + 4 {(- 4 \sin^{3} (x))}^{3} (3 \sin (x)) + 6 {(- 4 \sin^{3} (x))}^{2} {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.3.2

Multiplique $3$ por $4$ .

$256 \sin^{12} (x) + 4 {(- 4 \sin^{3} (x))}^{3} (3 \sin (x)) + 6 {(- 4 \sin^{3} (x))}^{2} {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.4

Aplique a regra do produto a $- 4 \sin^{3} (x)$ .

$256 \sin^{12} (x) + 4 ({(- 4)}^{3} {(\sin^{3} (x))}^{3}) (3 \sin (x)) + 6 {(- 4 \sin^{3} (x))}^{2} {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.5

Eleve $- 4$ à potência de $3$ .

$256 \sin^{12} (x) + 4 (- 64 {(\sin^{3} (x))}^{3}) (3 \sin (x)) + 6 {(- 4 \sin^{3} (x))}^{2} {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.6

Multiplique os expoentes em ${(\sin^{3} (x))}^{3}$ .

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Etapa 3.6.1

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

$256 \sin^{12} (x) + 4 (- 64 {\sin (x)}^{3 \cdot 3}) (3 \sin (x)) + 6 {(- 4 \sin^{3} (x))}^{2} {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.6.2

Multiplique $3$ por $3$ .

$256 \sin^{12} (x) + 4 (- 64 \sin^{9} (x)) (3 \sin (x)) + 6 {(- 4 \sin^{3} (x))}^{2} {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.7

Multiplique $\sin^{9} (x)$ por $\sin (x)$ somando os expoentes.

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Etapa 3.7.1

Mova $\sin (x)$ .

$256 \sin^{12} (x) + 4 (- 64 (\sin (x) \sin^{9} (x))) \cdot 3 + 6 {(- 4 \sin^{3} (x))}^{2} {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.7.2

Multiplique $\sin (x)$ por $\sin^{9} (x)$ .

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Etapa 3.7.2.1

Eleve $\sin (x)$ à potência de $1$ .

$256 \sin^{12} (x) + 4 (- 64 (\sin^{1} (x) \sin^{9} (x))) \cdot 3 + 6 {(- 4 \sin^{3} (x))}^{2} {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.7.2.2

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

$256 \sin^{12} (x) + 4 (- 64 {\sin (x)}^{1 + 9}) \cdot 3 + 6 {(- 4 \sin^{3} (x))}^{2} {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.7.3

Some $1$ e $9$ .

$256 \sin^{12} (x) + 4 (- 64 \sin^{10} (x)) \cdot 3 + 6 {(- 4 \sin^{3} (x))}^{2} {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.8

Multiplique $- 64$ por $4$ .

$256 \sin^{12} (x) - 256 \sin^{10} (x) \cdot 3 + 6 {(- 4 \sin^{3} (x))}^{2} {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.9

Multiplique $3$ por $- 256$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 6 {(- 4 \sin^{3} (x))}^{2} {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.10

Aplique a regra do produto a $- 4 \sin^{3} (x)$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 6 ({(- 4)}^{2} {(\sin^{3} (x))}^{2}) {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.11

Eleve $- 4$ à potência de $2$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 6 (16 {(\sin^{3} (x))}^{2}) {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.12

Multiplique os expoentes em ${(\sin^{3} (x))}^{2}$ .

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Etapa 3.12.1

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

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 6 (16 {\sin (x)}^{3 \cdot 2}) {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.12.2

Multiplique $3$ por $2$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 6 (16 \sin^{6} (x)) {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.13

Multiplique $16$ por $6$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 96 \sin^{6} (x) {(3 \sin (x))}^{2} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.14

Aplique a regra do produto a $3 \sin (x)$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 96 \sin^{6} (x) (3^{2} \sin^{2} (x)) + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.15

Reescreva usando a propriedade comutativa da multiplicação.

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 96 \cdot 3^{2} \sin^{6} (x) \sin^{2} (x) + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.16

Multiplique $\sin^{6} (x)$ por $\sin^{2} (x)$ somando os expoentes.

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Etapa 3.16.1

Mova $\sin^{2} (x)$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 96 \cdot 3^{2} (\sin^{2} (x) \sin^{6} (x)) + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.16.2

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

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 96 \cdot 3^{2} {\sin (x)}^{2 + 6} + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.16.3

Some $2$ e $6$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 96 \cdot 3^{2} \sin^{8} (x) + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.17

Eleve $3$ à potência de $2$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 96 \cdot 9 \sin^{8} (x) + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.18

Multiplique $96$ por $9$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 864 \sin^{8} (x) + 4 (- 4 \sin^{3} (x)) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.19

Multiplique $- 4$ por $4$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 864 \sin^{8} (x) - 16 \sin^{3} (x) {(3 \sin (x))}^{3} + {(3 \sin (x))}^{4}$

Etapa 3.20

Aplique a regra do produto a $3 \sin (x)$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 864 \sin^{8} (x) - 16 \sin^{3} (x) (3^{3} \sin^{3} (x)) + {(3 \sin (x))}^{4}$

Etapa 3.21

Reescreva usando a propriedade comutativa da multiplicação.

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 864 \sin^{8} (x) - 16 \cdot 3^{3} \sin^{3} (x) \sin^{3} (x) + {(3 \sin (x))}^{4}$

Etapa 3.22

Multiplique $\sin^{3} (x)$ por $\sin^{3} (x)$ somando os expoentes.

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Etapa 3.22.1

Mova $\sin^{3} (x)$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 864 \sin^{8} (x) - 16 \cdot 3^{3} (\sin^{3} (x) \sin^{3} (x)) + {(3 \sin (x))}^{4}$

Etapa 3.22.2

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

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 864 \sin^{8} (x) - 16 \cdot 3^{3} {\sin (x)}^{3 + 3} + {(3 \sin (x))}^{4}$

Etapa 3.22.3

Some $3$ e $3$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 864 \sin^{8} (x) - 16 \cdot 3^{3} \sin^{6} (x) + {(3 \sin (x))}^{4}$

Etapa 3.23

Eleve $3$ à potência de $3$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 864 \sin^{8} (x) - 16 \cdot 27 \sin^{6} (x) + {(3 \sin (x))}^{4}$

Etapa 3.24

Multiplique $- 16$ por $27$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 864 \sin^{8} (x) - 432 \sin^{6} (x) + {(3 \sin (x))}^{4}$

Etapa 3.25

Aplique a regra do produto a $3 \sin (x)$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 864 \sin^{8} (x) - 432 \sin^{6} (x) + 3^{4} \sin^{4} (x)$

Etapa 3.26

Eleve $3$ à potência de $4$ .

$256 \sin^{12} (x) - 768 \sin^{10} (x) + 864 \sin^{8} (x) - 432 \sin^{6} (x) + 81 \sin^{4} (x)$