Algebra Examples

${(1 + i)}^{5} {(1 - i)}^{5}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

One to any power is one.

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

Step 2.1.2

One to any power is one.

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

Step 2.1.3

Multiply $5$ by $1$ .

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

Step 2.1.4

One to any power is one.

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

Step 2.1.5

Multiply $10$ by $1$ .

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

Step 2.1.6

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

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

Step 2.1.7

Multiply $10$ by $- 1$ .

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

Step 2.1.8

One to any power is one.

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

Step 2.1.9

Multiply $10$ by $1$ .

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

Step 2.1.10

Factor out $i^{2}$ .

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

Step 2.1.11

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

$(1 + 5 i - 10 + 10 (- 1 \cdot i) + 5 \cdot 1 i^{4} + i^{5}) {(1 - i)}^{5}$

Step 2.1.12

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

$(1 + 5 i - 10 + 10 (- i) + 5 \cdot 1 i^{4} + i^{5}) {(1 - i)}^{5}$

Step 2.1.13

Multiply $- 1$ by $10$ .

$(1 + 5 i - 10 - 10 i + 5 \cdot 1 i^{4} + i^{5}) {(1 - i)}^{5}$

Step 2.1.14

Multiply $5$ by $1$ .

$(1 + 5 i - 10 - 10 i + 5 i^{4} + i^{5}) {(1 - i)}^{5}$

Step 2.1.15

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

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Step 2.1.15.1

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

$(1 + 5 i - 10 - 10 i + 5 {(i^{2})}^{2} + i^{5}) {(1 - i)}^{5}$

Step 2.1.15.2

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

$(1 + 5 i - 10 - 10 i + 5 {(- 1)}^{2} + i^{5}) {(1 - i)}^{5}$

Step 2.1.15.3

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

$(1 + 5 i - 10 - 10 i + 5 \cdot 1 + i^{5}) {(1 - i)}^{5}$

Step 2.1.16

Multiply $5$ by $1$ .

$(1 + 5 i - 10 - 10 i + 5 + i^{5}) {(1 - i)}^{5}$

Step 2.1.17

Factor out $i^{4}$ .

$(1 + 5 i - 10 - 10 i + 5 + i^{4} i) {(1 - i)}^{5}$

Step 2.1.18

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

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Step 2.1.18.1

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

$(1 + 5 i - 10 - 10 i + 5 + {(i^{2})}^{2} i) {(1 - i)}^{5}$

Step 2.1.18.2

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

$(1 + 5 i - 10 - 10 i + 5 + {(- 1)}^{2} i) {(1 - i)}^{5}$

Step 2.1.18.3

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

$(1 + 5 i - 10 - 10 i + 5 + 1 i) {(1 - i)}^{5}$

Step 2.1.19

Multiply $i$ by $1$ .

$(1 + 5 i - 10 - 10 i + 5 + i) {(1 - i)}^{5}$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $10$ from $1$ .

$(- 9 + 5 i - 10 i + 5 + i) {(1 - i)}^{5}$

Step 2.2.2

Add $- 9$ and $5$ .

$(- 4 + 5 i - 10 i + i) {(1 - i)}^{5}$

Step 2.2.3

Subtract $10 i$ from $5 i$ .

$(- 4 - 5 i + i) {(1 - i)}^{5}$

Step 2.2.4

Add $- 5 i$ and $i$ .

$(- 4 - 4 i) {(1 - i)}^{5}$

Step 3

Use the Binomial Theorem.

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

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

One to any power is one.

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

Step 4.1.2

One to any power is one.

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

Step 4.1.3

Multiply $5$ by $1$ .

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

Step 4.1.4

Multiply $- 1$ by $5$ .

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

Step 4.1.5

One to any power is one.

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

Step 4.1.6

Multiply $10$ by $1$ .

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

Step 4.1.7

Apply the product rule to $- i$ .

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

Step 4.1.8

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

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

Step 4.1.9

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

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

Step 4.1.10

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

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

Step 4.1.11

Multiply $10$ by $- 1$ .

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

Step 4.1.12

One to any power is one.

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

Step 4.1.13

Multiply $10$ by $1$ .

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

Step 4.1.14

Apply the product rule to $- i$ .

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

Step 4.1.15

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

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

Step 4.1.16

Factor out $i^{2}$ .

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

Step 4.1.17

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

$(- 4 - 4 i) (1 - 5 i - 10 + 10 (- (- 1 \cdot i)) + 5 \cdot 1 {(- i)}^{4} + {(- i)}^{5})$

Step 4.1.18

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

$(- 4 - 4 i) (1 - 5 i - 10 + 10 (- - i) + 5 \cdot 1 {(- i)}^{4} + {(- i)}^{5})$

Step 4.1.19

Multiply $- 1$ by $- 1$ .

$(- 4 - 4 i) (1 - 5 i - 10 + 10 (1 i) + 5 \cdot 1 {(- i)}^{4} + {(- i)}^{5})$

Step 4.1.20

Multiply $i$ by $1$ .

$(- 4 - 4 i) (1 - 5 i - 10 + 10 i + 5 \cdot 1 {(- i)}^{4} + {(- i)}^{5})$

Step 4.1.21

Multiply $5$ by $1$ .

$(- 4 - 4 i) (1 - 5 i - 10 + 10 i + 5 {(- i)}^{4} + {(- i)}^{5})$

Step 4.1.22

Apply the product rule to $- i$ .

$(- 4 - 4 i) (1 - 5 i - 10 + 10 i + 5 ({(- 1)}^{4} i^{4}) + {(- i)}^{5})$

Step 4.1.23

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

$(- 4 - 4 i) (1 - 5 i - 10 + 10 i + 5 (1 i^{4}) + {(- i)}^{5})$

Step 4.1.24

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

$(- 4 - 4 i) (1 - 5 i - 10 + 10 i + 5 i^{4} + {(- i)}^{5})$

Step 4.1.25

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

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Step 4.1.25.1

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

$(- 4 - 4 i) (1 - 5 i - 10 + 10 i + 5 {(i^{2})}^{2} + {(- i)}^{5})$

Step 4.1.25.2

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

$(- 4 - 4 i) (1 - 5 i - 10 + 10 i + 5 {(- 1)}^{2} + {(- i)}^{5})$

Step 4.1.25.3

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

$(- 4 - 4 i) (1 - 5 i - 10 + 10 i + 5 \cdot 1 + {(- i)}^{5})$

Step 4.1.26

Multiply $5$ by $1$ .

$(- 4 - 4 i) (1 - 5 i - 10 + 10 i + 5 + {(- i)}^{5})$

Step 4.1.27

Apply the product rule to $- i$ .

$(- 4 - 4 i) (1 - 5 i - 10 + 10 i + 5 + {(- 1)}^{5} i^{5})$

Step 4.1.28

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

$(- 4 - 4 i) (1 - 5 i - 10 + 10 i + 5 - i^{5})$

Step 4.1.29

Factor out $i^{4}$ .

$(- 4 - 4 i) (1 - 5 i - 10 + 10 i + 5 - (i^{4} i))$

Step 4.1.30

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

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Step 4.1.30.1

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

$(- 4 - 4 i) (1 - 5 i - 10 + 10 i + 5 - ({(i^{2})}^{2} i))$

Step 4.1.30.2

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

$(- 4 - 4 i) (1 - 5 i - 10 + 10 i + 5 - ({(- 1)}^{2} i))$

Step 4.1.30.3

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

$(- 4 - 4 i) (1 - 5 i - 10 + 10 i + 5 - (1 i))$

Step 4.1.31

Multiply $i$ by $1$ .

$(- 4 - 4 i) (1 - 5 i - 10 + 10 i + 5 - i)$

Step 4.2

Simplify by adding terms.

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Step 4.2.1

Subtract $10$ from $1$ .

$(- 4 - 4 i) (- 9 - 5 i + 10 i + 5 - i)$

Step 4.2.2

Add $- 9$ and $5$ .

$(- 4 - 4 i) (- 4 - 5 i + 10 i - i)$

Step 4.2.3

Add $- 5 i$ and $10 i$ .

$(- 4 - 4 i) (- 4 + 5 i - i)$

Step 4.2.4

Subtract $i$ from $5 i$ .

$(- 4 - 4 i) (- 4 + 4 i)$

Step 5

Expand $(- 4 - 4 i) (- 4 + 4 i)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$- 4 (- 4 + 4 i) - 4 i (- 4 + 4 i)$

Step 5.2

Apply the distributive property.

$- 4 \cdot - 4 - 4 (4 i) - 4 i (- 4 + 4 i)$

Step 5.3

Apply the distributive property.

$- 4 \cdot - 4 - 4 (4 i) - 4 i \cdot - 4 - 4 i (4 i)$

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 $- 4$ by $- 4$ .

$16 - 4 (4 i) - 4 i \cdot - 4 - 4 i (4 i)$

Step 6.1.2

Multiply $4$ by $- 4$ .

$16 - 16 i - 4 i \cdot - 4 - 4 i (4 i)$

Step 6.1.3

Multiply $- 4$ by $- 4$ .

$16 - 16 i + 16 i - 4 i (4 i)$

Step 6.1.4

Multiply $- 4 i (4 i)$ .

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Step 6.1.4.1

Multiply $4$ by $- 4$ .

$16 - 16 i + 16 i - 16 i i$

Step 6.1.4.2

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

$16 - 16 i + 16 i - 16 (i^{1} i)$

Step 6.1.4.3

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

$16 - 16 i + 16 i - 16 (i^{1} i^{1})$

Step 6.1.4.4

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

$16 - 16 i + 16 i - 16 i^{1 + 1}$

Step 6.1.4.5

Add $1$ and $1$ .

$16 - 16 i + 16 i - 16 i^{2}$

Step 6.1.5

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

$16 - 16 i + 16 i - 16 \cdot - 1$

Step 6.1.6

Multiply $- 16$ by $- 1$ .

$16 - 16 i + 16 i + 16$

Step 6.2

Add $16$ and $16$ .

$32 - 16 i + 16 i$

Step 6.3

Add $- 16 i$ and $16 i$ .

$32 + 0$

Step 6.4

Add $32$ and $0$ .

$32$