Simplify 5i(4-i)^2

$5 i {(4 - i)}^{2}$

Step 1

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

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

Step 2

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$5 i (16 + 4 (- i) - i \cdot 4 - i (- i))$

Multiply $- 1$ by $4$ .

$5 i (16 - 4 i - i \cdot 4 - i (- i))$

Multiply $4$ by $- 1$ .

$5 i (16 - 4 i - 4 i - i (- i))$

Multiply $- i (- i)$ .

Multiply $- 1$ by $- 1$ .

$5 i (16 - 4 i - 4 i + 1 i i)$

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

$5 i (16 - 4 i - 4 i + 1 (i^{1} i))$

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

$5 i (16 - 4 i - 4 i + 1 (i^{1} i^{1}))$

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

$5 i (16 - 4 i - 4 i + 1 i^{1 + 1})$

Add $1$ and $1$ .

$5 i (16 - 4 i - 4 i + 1 i^{2})$

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

$5 i (16 - 4 i - 4 i + i^{2})$

$5 i (16 - 4 i - 4 i + i^{2})$

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

$5 i (16 - 4 i - 4 i - 1)$

$5 i (16 - 4 i - 4 i - 1)$

Subtract $1$ from $16$ .

$5 i (15 - 4 i - 4 i)$

Subtract $4 i$ from $- 4 i$ .

$5 i (15 - 8 i)$

$5 i (15 - 8 i)$

Step 4

Apply the distributive property.

$5 i \cdot 15 + 5 i (- 8 i)$

Step 5

Multiply $15$ by $5$ .

$75 i + 5 i (- 8 i)$

Step 6

Multiply $5 i (- 8 i)$ .

Multiply $- 8$ by $5$ .

$75 i - 40 i i$

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

$75 i - 40 (i^{1} i)$

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

$75 i - 40 (i^{1} i^{1})$

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

$75 i - 40 i^{1 + 1}$

Add $1$ and $1$ .

$75 i - 40 i^{2}$

$75 i - 40 i^{2}$

Step 7

Simplify each term.

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

$75 i - 40 \cdot - 1$

Multiply $- 40$ by $- 1$ .

$75 i + 40$

$75 i + 40$

Step 8

Reorder $75 i$ and $40$ .

$40 + 75 i$

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