Simplify (2i)(1-4i)(1+i)

$(2 i) (1 - 4 i) (1 + i)$

Step 1

Apply the distributive property.

$(2 i \cdot 1 + 2 i (- 4 i)) (1 + i)$

Step 2

Multiply $2$ by $1$ .

$(2 i + 2 i (- 4 i)) (1 + i)$

Step 3

Multiply $2 i (- 4 i)$ .

Multiply $- 4$ by $2$ .

$(2 i - 8 i i) (1 + i)$

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

$(2 i - 8 (i^{1} i)) (1 + i)$

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

$(2 i - 8 (i^{1} i^{1})) (1 + i)$

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

$(2 i - 8 i^{1 + 1}) (1 + i)$

Add $1$ and $1$ .

$(2 i - 8 i^{2}) (1 + i)$

Step 4

Simplify each term.

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

$(2 i - 8 \cdot - 1) (1 + i)$

Multiply $- 8$ by $- 1$ .

$(2 i + 8) (1 + i)$

Step 5

Expand $(2 i + 8) (1 + i)$ using the FOIL Method.

Apply the distributive property.

$2 i (1 + i) + 8 (1 + i)$

Apply the distributive property.

$2 i \cdot 1 + 2 i i + 8 (1 + i)$

Apply the distributive property.

$2 i \cdot 1 + 2 i i + 8 \cdot 1 + 8 i$

Step 6

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $1$ .

$2 i + 2 i i + 8 \cdot 1 + 8 i$

Multiply $2 i i$ .

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

$2 i + 2 (i^{1} i) + 8 \cdot 1 + 8 i$

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

$2 i + 2 (i^{1} i^{1}) + 8 \cdot 1 + 8 i$

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

$2 i + 2 i^{1 + 1} + 8 \cdot 1 + 8 i$

Add $1$ and $1$ .

$2 i + 2 i^{2} + 8 \cdot 1 + 8 i$

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

$2 i + 2 \cdot - 1 + 8 \cdot 1 + 8 i$

Multiply $2$ by $- 1$ .

$2 i - 2 + 8 \cdot 1 + 8 i$

Multiply $8$ by $1$ .

$2 i - 2 + 8 + 8 i$

Add $2 i$ and $8 i$ .

$- 2 + 8 + 10 i$

Add $- 2$ and $8$ .

$6 + 10 i$

Simplify (2i)(1-4i)(1+i)

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