Trigonometry Examples

${(5 - i)}^{6}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $5$ to the power of $6$ .

$15625 + 6 \cdot 5^{5} (- i) + 15 \cdot 5^{4} {(- i)}^{2} + 20 \cdot 5^{3} {(- i)}^{3} + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.2

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

$15625 + 6 \cdot 3125 (- i) + 15 \cdot 5^{4} {(- i)}^{2} + 20 \cdot 5^{3} {(- i)}^{3} + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.3

Multiply $6$ by $3125$ .

$15625 + 18750 (- i) + 15 \cdot 5^{4} {(- i)}^{2} + 20 \cdot 5^{3} {(- i)}^{3} + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.4

Multiply $- 1$ by $18750$ .

$15625 - 18750 i + 15 \cdot 5^{4} {(- i)}^{2} + 20 \cdot 5^{3} {(- i)}^{3} + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.5

Raise $5$ to the power of $4$ .

$15625 - 18750 i + 15 \cdot 625 {(- i)}^{2} + 20 \cdot 5^{3} {(- i)}^{3} + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.6

Multiply $15$ by $625$ .

$15625 - 18750 i + 9375 {(- i)}^{2} + 20 \cdot 5^{3} {(- i)}^{3} + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.7

Apply the product rule to $- i$ .

$15625 - 18750 i + 9375 ({(- 1)}^{2} i^{2}) + 20 \cdot 5^{3} {(- i)}^{3} + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.8

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

$15625 - 18750 i + 9375 (1 i^{2}) + 20 \cdot 5^{3} {(- i)}^{3} + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.9

Multiply $i^{2}$ by $1$ .

$15625 - 18750 i + 9375 i^{2} + 20 \cdot 5^{3} {(- i)}^{3} + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.10

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

$15625 - 18750 i + 9375 \cdot - 1 + 20 \cdot 5^{3} {(- i)}^{3} + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.11

Multiply $9375$ by $- 1$ .

$15625 - 18750 i - 9375 + 20 \cdot 5^{3} {(- i)}^{3} + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.12

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

$15625 - 18750 i - 9375 + 20 \cdot 125 {(- i)}^{3} + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.13

Multiply $20$ by $125$ .

$15625 - 18750 i - 9375 + 2500 {(- i)}^{3} + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.14

Apply the product rule to $- i$ .

$15625 - 18750 i - 9375 + 2500 ({(- 1)}^{3} i^{3}) + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.15

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

$15625 - 18750 i - 9375 + 2500 (- i^{3}) + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.16

Factor out $i^{2}$ .

$15625 - 18750 i - 9375 + 2500 (- (i^{2} \cdot i)) + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.17

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

$15625 - 18750 i - 9375 + 2500 (- (- 1 \cdot i)) + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.18

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

$15625 - 18750 i - 9375 + 2500 (- - i) + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.19

Multiply $- 1$ by $- 1$ .

$15625 - 18750 i - 9375 + 2500 (1 i) + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.20

Multiply $i$ by $1$ .

$15625 - 18750 i - 9375 + 2500 i + 15 \cdot 5^{2} {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.21

Raise $5$ to the power of $2$ .

$15625 - 18750 i - 9375 + 2500 i + 15 \cdot 25 {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.22

Multiply $15$ by $25$ .

$15625 - 18750 i - 9375 + 2500 i + 375 {(- i)}^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.23

Apply the product rule to $- i$ .

$15625 - 18750 i - 9375 + 2500 i + 375 ({(- 1)}^{4} i^{4}) + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.24

Raise $- 1$ to the power of $4$ .

$15625 - 18750 i - 9375 + 2500 i + 375 (1 i^{4}) + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.25

Multiply $i^{4}$ by $1$ .

$15625 - 18750 i - 9375 + 2500 i + 375 i^{4} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.26

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

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Step 2.1.26.1

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

$15625 - 18750 i - 9375 + 2500 i + 375 {(i^{2})}^{2} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.26.2

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

$15625 - 18750 i - 9375 + 2500 i + 375 {(- 1)}^{2} + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.26.3

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

$15625 - 18750 i - 9375 + 2500 i + 375 \cdot 1 + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.27

Multiply $375$ by $1$ .

$15625 - 18750 i - 9375 + 2500 i + 375 + 6 \cdot 5 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.28

Multiply $6$ by $5$ .

$15625 - 18750 i - 9375 + 2500 i + 375 + 30 {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.29

Apply the product rule to $- i$ .

$15625 - 18750 i - 9375 + 2500 i + 375 + 30 ({(- 1)}^{5} i^{5}) + {(- i)}^{6}$

Step 2.1.30

Raise $- 1$ to the power of $5$ .

$15625 - 18750 i - 9375 + 2500 i + 375 + 30 (- i^{5}) + {(- i)}^{6}$

Step 2.1.31

Factor out $i^{4}$ .

$15625 - 18750 i - 9375 + 2500 i + 375 + 30 (- (i^{4} i)) + {(- i)}^{6}$

Step 2.1.32

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

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Step 2.1.32.1

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

$15625 - 18750 i - 9375 + 2500 i + 375 + 30 (- ({(i^{2})}^{2} i)) + {(- i)}^{6}$

Step 2.1.32.2

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

$15625 - 18750 i - 9375 + 2500 i + 375 + 30 (- ({(- 1)}^{2} i)) + {(- i)}^{6}$

Step 2.1.32.3

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

$15625 - 18750 i - 9375 + 2500 i + 375 + 30 (- (1 i)) + {(- i)}^{6}$

Step 2.1.33

Multiply $i$ by $1$ .

$15625 - 18750 i - 9375 + 2500 i + 375 + 30 (- i) + {(- i)}^{6}$

Step 2.1.34

Multiply $- 1$ by $30$ .

$15625 - 18750 i - 9375 + 2500 i + 375 - 30 i + {(- i)}^{6}$

Step 2.1.35

Apply the product rule to $- i$ .

$15625 - 18750 i - 9375 + 2500 i + 375 - 30 i + {(- 1)}^{6} i^{6}$

Step 2.1.36

Raise $- 1$ to the power of $6$ .

$15625 - 18750 i - 9375 + 2500 i + 375 - 30 i + 1 i^{6}$

Step 2.1.37

Multiply $i^{6}$ by $1$ .

$15625 - 18750 i - 9375 + 2500 i + 375 - 30 i + i^{6}$

Step 2.1.38

Factor out $i^{4}$ .

$15625 - 18750 i - 9375 + 2500 i + 375 - 30 i + i^{4} i^{2}$

Step 2.1.39

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

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Step 2.1.39.1

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

$15625 - 18750 i - 9375 + 2500 i + 375 - 30 i + {(i^{2})}^{2} i^{2}$

Step 2.1.39.2

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

$15625 - 18750 i - 9375 + 2500 i + 375 - 30 i + {(- 1)}^{2} i^{2}$

Step 2.1.39.3

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

$15625 - 18750 i - 9375 + 2500 i + 375 - 30 i + 1 i^{2}$

Step 2.1.40

Multiply $i^{2}$ by $1$ .

$15625 - 18750 i - 9375 + 2500 i + 375 - 30 i + i^{2}$

Step 2.1.41

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

$15625 - 18750 i - 9375 + 2500 i + 375 - 30 i - 1$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $9375$ from $15625$ .

$6250 - 18750 i + 2500 i + 375 - 30 i - 1$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $6250$ and $375$ .

$6625 - 18750 i + 2500 i - 30 i - 1$

Step 2.2.2.2

Subtract $1$ from $6625$ .

$6624 - 18750 i + 2500 i - 30 i$

Step 2.2.3

Add $- 18750 i$ and $2500 i$ .

$6624 - 16250 i - 30 i$

Step 2.2.4

Subtract $30 i$ from $- 16250 i$ .

$6624 - 16280 i$