Convert to Rectangular (4-4i)^6

${(4 - 4 i)}^{6}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

Simplify each term.

Raise $4$ to the power of $6$ .

$4096 + 6 \cdot 4^{5} (- 4 i) + 15 \cdot 4^{4} {(- 4 i)}^{2} + 20 \cdot 4^{3} {(- 4 i)}^{3} + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

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

$4096 + 6 \cdot 1024 (- 4 i) + 15 \cdot 4^{4} {(- 4 i)}^{2} + 20 \cdot 4^{3} {(- 4 i)}^{3} + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Multiply $6$ by $1024$ .

$4096 + 6144 (- 4 i) + 15 \cdot 4^{4} {(- 4 i)}^{2} + 20 \cdot 4^{3} {(- 4 i)}^{3} + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Multiply $- 4$ by $6144$ .

$4096 - 24576 i + 15 \cdot 4^{4} {(- 4 i)}^{2} + 20 \cdot 4^{3} {(- 4 i)}^{3} + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

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

$4096 - 24576 i + 15 \cdot 256 {(- 4 i)}^{2} + 20 \cdot 4^{3} {(- 4 i)}^{3} + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Multiply $15$ by $256$ .

$4096 - 24576 i + 3840 {(- 4 i)}^{2} + 20 \cdot 4^{3} {(- 4 i)}^{3} + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Apply the product rule to $- 4 i$ .

$4096 - 24576 i + 3840 ({(- 4)}^{2} i^{2}) + 20 \cdot 4^{3} {(- 4 i)}^{3} + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Raise $- 4$ to the power of $2$ .

$4096 - 24576 i + 3840 (16 i^{2}) + 20 \cdot 4^{3} {(- 4 i)}^{3} + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

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

$4096 - 24576 i + 3840 (16 \cdot - 1) + 20 \cdot 4^{3} {(- 4 i)}^{3} + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Multiply $3840 (16 \cdot - 1)$ .

Multiply $16$ by $- 1$ .

$4096 - 24576 i + 3840 \cdot - 16 + 20 \cdot 4^{3} {(- 4 i)}^{3} + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Multiply $3840$ by $- 16$ .

$4096 - 24576 i - 61440 + 20 \cdot 4^{3} {(- 4 i)}^{3} + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

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

$4096 - 24576 i - 61440 + 20 \cdot 64 {(- 4 i)}^{3} + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Multiply $20$ by $64$ .

$4096 - 24576 i - 61440 + 1280 {(- 4 i)}^{3} + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Apply the product rule to $- 4 i$ .

$4096 - 24576 i - 61440 + 1280 ({(- 4)}^{3} i^{3}) + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

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

$4096 - 24576 i - 61440 + 1280 (- 64 i^{3}) + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Factor out $i^{2}$ .

$4096 - 24576 i - 61440 + 1280 (- 64 (i^{2} \cdot i)) + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

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

$4096 - 24576 i - 61440 + 1280 (- 64 (- 1 \cdot i)) + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

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

$4096 - 24576 i - 61440 + 1280 (- 64 (- i)) + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Multiply $- 1$ by $- 64$ .

$4096 - 24576 i - 61440 + 1280 (64 i) + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Multiply $64$ by $1280$ .

$4096 - 24576 i - 61440 + 81920 i + 15 \cdot 4^{2} {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Raise $4$ to the power of $2$ .

$4096 - 24576 i - 61440 + 81920 i + 15 \cdot 16 {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Multiply $15$ by $16$ .

$4096 - 24576 i - 61440 + 81920 i + 240 {(- 4 i)}^{4} + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Apply the product rule to $- 4 i$ .

$4096 - 24576 i - 61440 + 81920 i + 240 ({(- 4)}^{4} i^{4}) + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

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

$4096 - 24576 i - 61440 + 81920 i + 240 (256 i^{4}) + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

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

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

$4096 - 24576 i - 61440 + 81920 i + 240 (256 {(i^{2})}^{2}) + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

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

$4096 - 24576 i - 61440 + 81920 i + 240 (256 {(- 1)}^{2}) + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

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

$4096 - 24576 i - 61440 + 81920 i + 240 (256 \cdot 1) + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Multiply $240 (256 \cdot 1)$ .

Multiply $256$ by $1$ .

$4096 - 24576 i - 61440 + 81920 i + 240 \cdot 256 + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Multiply $240$ by $256$ .

$4096 - 24576 i - 61440 + 81920 i + 61440 + 6 \cdot 4 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Multiply $6$ by $4$ .

$4096 - 24576 i - 61440 + 81920 i + 61440 + 24 {(- 4 i)}^{5} + {(- 4 i)}^{6}$

Apply the product rule to $- 4 i$ .

$4096 - 24576 i - 61440 + 81920 i + 61440 + 24 ({(- 4)}^{5} i^{5}) + {(- 4 i)}^{6}$

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

$4096 - 24576 i - 61440 + 81920 i + 61440 + 24 (- 1024 i^{5}) + {(- 4 i)}^{6}$

Factor out $i^{4}$ .

$4096 - 24576 i - 61440 + 81920 i + 61440 + 24 (- 1024 (i^{4} i)) + {(- 4 i)}^{6}$

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

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

$4096 - 24576 i - 61440 + 81920 i + 61440 + 24 (- 1024 ({(i^{2})}^{2} i)) + {(- 4 i)}^{6}$

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

$4096 - 24576 i - 61440 + 81920 i + 61440 + 24 (- 1024 ({(- 1)}^{2} i)) + {(- 4 i)}^{6}$

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

$4096 - 24576 i - 61440 + 81920 i + 61440 + 24 (- 1024 (1 i)) + {(- 4 i)}^{6}$

Multiply $i$ by $1$ .

$4096 - 24576 i - 61440 + 81920 i + 61440 + 24 (- 1024 i) + {(- 4 i)}^{6}$

Multiply $- 1024$ by $24$ .

$4096 - 24576 i - 61440 + 81920 i + 61440 - 24576 i + {(- 4 i)}^{6}$

Apply the product rule to $- 4 i$ .

$4096 - 24576 i - 61440 + 81920 i + 61440 - 24576 i + {(- 4)}^{6} i^{6}$

Raise $- 4$ to the power of $6$ .

$4096 - 24576 i - 61440 + 81920 i + 61440 - 24576 i + 4096 i^{6}$

Factor out $i^{4}$ .

$4096 - 24576 i - 61440 + 81920 i + 61440 - 24576 i + 4096 (i^{4} i^{2})$

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

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

$4096 - 24576 i - 61440 + 81920 i + 61440 - 24576 i + 4096 ({(i^{2})}^{2} i^{2})$

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

$4096 - 24576 i - 61440 + 81920 i + 61440 - 24576 i + 4096 ({(- 1)}^{2} i^{2})$

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

$4096 - 24576 i - 61440 + 81920 i + 61440 - 24576 i + 4096 (1 i^{2})$

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

$4096 - 24576 i - 61440 + 81920 i + 61440 - 24576 i + 4096 i^{2}$

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

$4096 - 24576 i - 61440 + 81920 i + 61440 - 24576 i + 4096 \cdot - 1$

Multiply $4096$ by $- 1$ .

$4096 - 24576 i - 61440 + 81920 i + 61440 - 24576 i - 4096$

Simplify by adding terms.

Subtract $61440$ from $4096$ .

$- 57344 - 24576 i + 81920 i + 61440 - 24576 i - 4096$

Simplify by adding and subtracting.

Add $- 57344$ and $61440$ .

$4096 - 24576 i + 81920 i - 24576 i - 4096$

Subtract $4096$ from $4096$ .

$0 - 24576 i + 81920 i - 24576 i$

Subtract $24576 i$ from $0$ .

$- 24576 i + 81920 i - 24576 i$

Add $- 24576 i$ and $81920 i$ .

$57344 i - 24576 i$

Subtract $24576 i$ from $57344 i$ .

$32768 i$

Convert to Rectangular (4-4i)^6

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