Trigonometry Examples

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $3$ to the power of $5$ .

$\frac{(243 + 5 \cdot 3^{4} (3 i) + 10 \cdot 3^{3} {(3 i)}^{2} + 10 \cdot 3^{2} {(3 i)}^{3} + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.2

Multiply $3^{4}$ by $3$ by adding the exponents.

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Step 2.1.2.1

Move $3$ .

$\frac{(243 + 5 \cdot (3 \cdot 3^{4}) i + 10 \cdot 3^{3} {(3 i)}^{2} + 10 \cdot 3^{2} {(3 i)}^{3} + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.2.2

Multiply $3$ by $3^{4}$ .

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Step 2.1.2.2.1

Raise $3$ to the power of $1$ .

$\frac{(243 + 5 \cdot (3^{1} \cdot 3^{4}) i + 10 \cdot 3^{3} {(3 i)}^{2} + 10 \cdot 3^{2} {(3 i)}^{3} + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(243 + 5 \cdot 3^{1 + 4} i + 10 \cdot 3^{3} {(3 i)}^{2} + 10 \cdot 3^{2} {(3 i)}^{3} + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.2.3

Add $1$ and $4$ .

$\frac{(243 + 5 \cdot 3^{5} i + 10 \cdot 3^{3} {(3 i)}^{2} + 10 \cdot 3^{2} {(3 i)}^{3} + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.3

Raise $3$ to the power of $5$ .

$\frac{(243 + 5 \cdot 243 i + 10 \cdot 3^{3} {(3 i)}^{2} + 10 \cdot 3^{2} {(3 i)}^{3} + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.4

Multiply $5$ by $243$ .

$\frac{(243 + 1215 i + 10 \cdot 3^{3} {(3 i)}^{2} + 10 \cdot 3^{2} {(3 i)}^{3} + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.5

Raise $3$ to the power of $3$ .

$\frac{(243 + 1215 i + 10 \cdot 27 {(3 i)}^{2} + 10 \cdot 3^{2} {(3 i)}^{3} + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.6

Multiply $10$ by $27$ .

$\frac{(243 + 1215 i + 270 {(3 i)}^{2} + 10 \cdot 3^{2} {(3 i)}^{3} + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.7

Apply the product rule to $3 i$ .

$\frac{(243 + 1215 i + 270 (3^{2} i^{2}) + 10 \cdot 3^{2} {(3 i)}^{3} + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.8

Raise $3$ to the power of $2$ .

$\frac{(243 + 1215 i + 270 (9 i^{2}) + 10 \cdot 3^{2} {(3 i)}^{3} + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.9

Rewrite $i^{2}$ as $- 1$ .

$\frac{(243 + 1215 i + 270 (9 \cdot - 1) + 10 \cdot 3^{2} {(3 i)}^{3} + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.10

Multiply $270 (9 \cdot - 1)$ .

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Step 2.1.10.1

Multiply $9$ by $- 1$ .

$\frac{(243 + 1215 i + 270 \cdot - 9 + 10 \cdot 3^{2} {(3 i)}^{3} + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.10.2

Multiply $270$ by $- 9$ .

$\frac{(243 + 1215 i - 2430 + 10 \cdot 3^{2} {(3 i)}^{3} + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.11

Raise $3$ to the power of $2$ .

$\frac{(243 + 1215 i - 2430 + 10 \cdot 9 {(3 i)}^{3} + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.12

Multiply $10$ by $9$ .

$\frac{(243 + 1215 i - 2430 + 90 {(3 i)}^{3} + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.13

Apply the product rule to $3 i$ .

$\frac{(243 + 1215 i - 2430 + 90 (3^{3} i^{3}) + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.14

Raise $3$ to the power of $3$ .

$\frac{(243 + 1215 i - 2430 + 90 (27 i^{3}) + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.15

Factor out $i^{2}$ .

$\frac{(243 + 1215 i - 2430 + 90 (27 (i^{2} \cdot i)) + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.16

Rewrite $i^{2}$ as $- 1$ .

$\frac{(243 + 1215 i - 2430 + 90 (27 (- 1 \cdot i)) + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.17

Rewrite $- 1 i$ as $- i$ .

$\frac{(243 + 1215 i - 2430 + 90 (27 (- i)) + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.18

Multiply $- 1$ by $27$ .

$\frac{(243 + 1215 i - 2430 + 90 (- 27 i) + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.19

Multiply $- 27$ by $90$ .

$\frac{(243 + 1215 i - 2430 - 2430 i + 5 \cdot 3 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.20

Multiply $5$ by $3$ .

$\frac{(243 + 1215 i - 2430 - 2430 i + 15 {(3 i)}^{4} + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.21

Apply the product rule to $3 i$ .

$\frac{(243 + 1215 i - 2430 - 2430 i + 15 (3^{4} i^{4}) + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.22

Raise $3$ to the power of $4$ .

$\frac{(243 + 1215 i - 2430 - 2430 i + 15 (81 i^{4}) + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.23

Rewrite $i^{4}$ as $1$ .

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Step 2.1.23.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$\frac{(243 + 1215 i - 2430 - 2430 i + 15 (81 {(i^{2})}^{2}) + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.23.2

Rewrite $i^{2}$ as $- 1$ .

$\frac{(243 + 1215 i - 2430 - 2430 i + 15 (81 {(- 1)}^{2}) + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.23.3

Raise $- 1$ to the power of $2$ .

$\frac{(243 + 1215 i - 2430 - 2430 i + 15 (81 \cdot 1) + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.24

Multiply $15 (81 \cdot 1)$ .

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Step 2.1.24.1

Multiply $81$ by $1$ .

$\frac{(243 + 1215 i - 2430 - 2430 i + 15 \cdot 81 + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.24.2

Multiply $15$ by $81$ .

$\frac{(243 + 1215 i - 2430 - 2430 i + 1215 + {(3 i)}^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.25

Apply the product rule to $3 i$ .

$\frac{(243 + 1215 i - 2430 - 2430 i + 1215 + 3^{5} i^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.26

Raise $3$ to the power of $5$ .

$\frac{(243 + 1215 i - 2430 - 2430 i + 1215 + 243 i^{5}) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.27

Factor out $i^{4}$ .

$\frac{(243 + 1215 i - 2430 - 2430 i + 1215 + 243 (i^{4} i)) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.28

Rewrite $i^{4}$ as $1$ .

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Step 2.1.28.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$\frac{(243 + 1215 i - 2430 - 2430 i + 1215 + 243 ({(i^{2})}^{2} i)) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.28.2

Rewrite $i^{2}$ as $- 1$ .

$\frac{(243 + 1215 i - 2430 - 2430 i + 1215 + 243 ({(- 1)}^{2} i)) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.28.3

Raise $- 1$ to the power of $2$ .

$\frac{(243 + 1215 i - 2430 - 2430 i + 1215 + 243 (1 i)) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.1.29

Multiply $i$ by $1$ .

$\frac{(243 + 1215 i - 2430 - 2430 i + 1215 + 243 i) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $2430$ from $243$ .

$\frac{(- 2187 + 1215 i - 2430 i + 1215 + 243 i) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.2.2

Add $- 2187$ and $1215$ .

$\frac{(- 972 + 1215 i - 2430 i + 243 i) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.2.3

Subtract $2430 i$ from $1215 i$ .

$\frac{(- 972 - 1215 i + 243 i) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 2.2.4

Add $- 1215 i$ and $243 i$ .

$\frac{(- 972 - 972 i) {(- 2 + 2 i)}^{3}}{{(\sqrt{3} + i)}^{10}}$

Step 3

Use the Binomial Theorem.

$\frac{(- 972 - 972 i) ({(- 2)}^{3} + 3 {(- 2)}^{2} (2 i) + 3 \cdot - 2 {(2 i)}^{2} + {(2 i)}^{3})}{{(\sqrt{3} + i)}^{10}}$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raise $- 2$ to the power of $3$ .

$\frac{(- 972 - 972 i) (- 8 + 3 {(- 2)}^{2} (2 i) + 3 \cdot - 2 {(2 i)}^{2} + {(2 i)}^{3})}{{(\sqrt{3} + i)}^{10}}$

Step 4.1.2

Raise $- 2$ to the power of $2$ .

$\frac{(- 972 - 972 i) (- 8 + 3 \cdot 4 (2 i) + 3 \cdot - 2 {(2 i)}^{2} + {(2 i)}^{3})}{{(\sqrt{3} + i)}^{10}}$

Step 4.1.3

Multiply $3$ by $4$ .

$\frac{(- 972 - 972 i) (- 8 + 12 (2 i) + 3 \cdot - 2 {(2 i)}^{2} + {(2 i)}^{3})}{{(\sqrt{3} + i)}^{10}}$

Step 4.1.4

Multiply $2$ by $12$ .

$\frac{(- 972 - 972 i) (- 8 + 24 i + 3 \cdot - 2 {(2 i)}^{2} + {(2 i)}^{3})}{{(\sqrt{3} + i)}^{10}}$

Step 4.1.5

Multiply $3$ by $- 2$ .

$\frac{(- 972 - 972 i) (- 8 + 24 i - 6 {(2 i)}^{2} + {(2 i)}^{3})}{{(\sqrt{3} + i)}^{10}}$

Step 4.1.6

Apply the product rule to $2 i$ .

$\frac{(- 972 - 972 i) (- 8 + 24 i - 6 (2^{2} i^{2}) + {(2 i)}^{3})}{{(\sqrt{3} + i)}^{10}}$

Step 4.1.7

Raise $2$ to the power of $2$ .

$\frac{(- 972 - 972 i) (- 8 + 24 i - 6 (4 i^{2}) + {(2 i)}^{3})}{{(\sqrt{3} + i)}^{10}}$

Step 4.1.8

Rewrite $i^{2}$ as $- 1$ .

$\frac{(- 972 - 972 i) (- 8 + 24 i - 6 (4 \cdot - 1) + {(2 i)}^{3})}{{(\sqrt{3} + i)}^{10}}$

Step 4.1.9

Multiply $- 6 (4 \cdot - 1)$ .

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Step 4.1.9.1

Multiply $4$ by $- 1$ .

$\frac{(- 972 - 972 i) (- 8 + 24 i - 6 \cdot - 4 + {(2 i)}^{3})}{{(\sqrt{3} + i)}^{10}}$

Step 4.1.9.2

Multiply $- 6$ by $- 4$ .

$\frac{(- 972 - 972 i) (- 8 + 24 i + 24 + {(2 i)}^{3})}{{(\sqrt{3} + i)}^{10}}$

Step 4.1.10

Apply the product rule to $2 i$ .

$\frac{(- 972 - 972 i) (- 8 + 24 i + 24 + 2^{3} i^{3})}{{(\sqrt{3} + i)}^{10}}$

Step 4.1.11

Raise $2$ to the power of $3$ .

$\frac{(- 972 - 972 i) (- 8 + 24 i + 24 + 8 i^{3})}{{(\sqrt{3} + i)}^{10}}$

Step 4.1.12

Factor out $i^{2}$ .

$\frac{(- 972 - 972 i) (- 8 + 24 i + 24 + 8 (i^{2} \cdot i))}{{(\sqrt{3} + i)}^{10}}$

Step 4.1.13

Rewrite $i^{2}$ as $- 1$ .

$\frac{(- 972 - 972 i) (- 8 + 24 i + 24 + 8 (- 1 \cdot i))}{{(\sqrt{3} + i)}^{10}}$

Step 4.1.14

Rewrite $- 1 i$ as $- i$ .

$\frac{(- 972 - 972 i) (- 8 + 24 i + 24 + 8 (- i))}{{(\sqrt{3} + i)}^{10}}$

Step 4.1.15

Multiply $- 1$ by $8$ .

$\frac{(- 972 - 972 i) (- 8 + 24 i + 24 - 8 i)}{{(\sqrt{3} + i)}^{10}}$

Step 4.2

Simplify by adding terms.

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Step 4.2.1

Add $- 8$ and $24$ .

$\frac{(- 972 - 972 i) (16 + 24 i - 8 i)}{{(\sqrt{3} + i)}^{10}}$

Step 4.2.2

Subtract $8 i$ from $24 i$ .

$\frac{(- 972 - 972 i) (16 + 16 i)}{{(\sqrt{3} + i)}^{10}}$

Step 5

Expand $(- 972 - 972 i) (16 + 16 i)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{- 972 (16 + 16 i) - 972 i (16 + 16 i)}{{(\sqrt{3} + i)}^{10}}$

Step 5.2

Apply the distributive property.

$\frac{- 972 \cdot 16 - 972 (16 i) - 972 i (16 + 16 i)}{{(\sqrt{3} + i)}^{10}}$

Step 5.3

Apply the distributive property.

$\frac{- 972 \cdot 16 - 972 (16 i) - 972 i \cdot 16 - 972 i (16 i)}{{(\sqrt{3} + i)}^{10}}$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $- 972$ by $16$ .

$\frac{- 15552 - 972 (16 i) - 972 i \cdot 16 - 972 i (16 i)}{{(\sqrt{3} + i)}^{10}}$

Step 6.1.2

Multiply $16$ by $- 972$ .

$\frac{- 15552 - 15552 i - 972 i \cdot 16 - 972 i (16 i)}{{(\sqrt{3} + i)}^{10}}$

Step 6.1.3

Multiply $16$ by $- 972$ .

$\frac{- 15552 - 15552 i - 15552 i - 972 i (16 i)}{{(\sqrt{3} + i)}^{10}}$

Step 6.1.4

Multiply $- 972 i (16 i)$ .

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Step 6.1.4.1

Multiply $16$ by $- 972$ .

$\frac{- 15552 - 15552 i - 15552 i - 15552 i i}{{(\sqrt{3} + i)}^{10}}$

Step 6.1.4.2

Raise $i$ to the power of $1$ .

$\frac{- 15552 - 15552 i - 15552 i - 15552 (i^{1} i)}{{(\sqrt{3} + i)}^{10}}$

Step 6.1.4.3

Raise $i$ to the power of $1$ .

$\frac{- 15552 - 15552 i - 15552 i - 15552 (i^{1} i^{1})}{{(\sqrt{3} + i)}^{10}}$

Step 6.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- 15552 - 15552 i - 15552 i - 15552 i^{1 + 1}}{{(\sqrt{3} + i)}^{10}}$

Step 6.1.4.5

Add $1$ and $1$ .

$\frac{- 15552 - 15552 i - 15552 i - 15552 i^{2}}{{(\sqrt{3} + i)}^{10}}$

Step 6.1.5

Rewrite $i^{2}$ as $- 1$ .

$\frac{- 15552 - 15552 i - 15552 i - 15552 \cdot - 1}{{(\sqrt{3} + i)}^{10}}$

Step 6.1.6

Multiply $- 15552$ by $- 1$ .

$\frac{- 15552 - 15552 i - 15552 i + 15552}{{(\sqrt{3} + i)}^{10}}$

Step 6.2

Add $- 15552$ and $15552$ .

$\frac{0 - 15552 i - 15552 i}{{(\sqrt{3} + i)}^{10}}$

Step 6.3

Subtract $15552 i$ from $0$ .

$\frac{- 15552 i - 15552 i}{{(\sqrt{3} + i)}^{10}}$

Step 6.4

Subtract $15552 i$ from $- 15552 i$ .

$\frac{- 31104 i}{{(\sqrt{3} + i)}^{10}}$

Step 7

Move the negative in front of the fraction.

$- \frac{31104 i}{{(\sqrt{3} + i)}^{10}}$