Simplify (1+i)^5(1-i)^5

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

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.

Simplify each term.

One to any power is one.

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

One to any power is one.

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

Multiply $5$ by $1$ .

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

One to any power is one.

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

Multiply $10$ by $1$ .

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

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

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

Multiply $10$ by $- 1$ .

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

One to any power is one.

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

Multiply $10$ by $1$ .

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

Factor out $i^{2}$ .

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

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

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

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

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

Multiply $- 1$ by $10$ .

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

Multiply $5$ by $1$ .

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

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

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

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

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

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

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

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

Multiply $5$ by $1$ .

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

Factor out $i^{4}$ .

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

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

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

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

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

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

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

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

Multiply $i$ by $1$ .

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

Simplify by adding terms.

Subtract $10$ from $1$ .

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

Add $- 9$ and $5$ .

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

Subtract $10 i$ from $5 i$ .

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

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.

Simplify each term.

One to any power is one.

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

One to any power is one.

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

Multiply $5$ by $1$ .

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

Multiply $- 1$ by $5$ .

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

One to any power is one.

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

Multiply $10$ by $1$ .

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

Apply the product rule to $- i$ .

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

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

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

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

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

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

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

Multiply $10$ by $- 1$ .

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

One to any power is one.

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

Multiply $10$ by $1$ .

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

Apply the product rule to $- i$ .

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

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

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

Factor out $i^{2}$ .

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

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

Multiply $i$ by $1$ .

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

Multiply $5$ by $1$ .

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

Apply the product rule to $- i$ .

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

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

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

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

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

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

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

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

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

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

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

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

Multiply $5$ by $1$ .

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

Apply the product rule to $- i$ .

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

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

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

Factor out $i^{4}$ .

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

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

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

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

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

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

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

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

Multiply $i$ by $1$ .

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

Simplify by adding terms.

Subtract $10$ from $1$ .

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

Add $- 9$ and $5$ .

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

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

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

Subtract $i$ from $5 i$ .

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

Step 5

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

Apply the distributive property.

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

Apply the distributive property.

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

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.

Simplify each term.

Multiply $- 4$ by $- 4$ .

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

Multiply $4$ by $- 4$ .

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

Multiply $- 4$ by $- 4$ .

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

Multiply $- 4 i (4 i)$ .

Multiply $4$ by $- 4$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 16$ by $- 1$ .

$16 - 16 i + 16 i + 16$

Add $16$ and $16$ .

$32 - 16 i + 16 i$

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

$32 + 0$

Add $32$ and $0$ .

$32$

Simplify (1+i)^5(1-i)^5

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