Simplify 4i(5-i)^2

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

Step 1

Rewrite ${(5 - i)}^{2}$ as $(5 - i) (5 - i)$ .

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

Step 2

Expand $(5 - i) (5 - i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$4 i (25 + 5 (- i) - i \cdot 5 - i (- i))$

Multiply $- 1$ by $5$ .

$4 i (25 - 5 i - i \cdot 5 - i (- i))$

Multiply $5$ by $- 1$ .

$4 i (25 - 5 i - 5 i - i (- i))$

Multiply $- i (- i)$ .

Multiply $- 1$ by $- 1$ .

$4 i (25 - 5 i - 5 i + 1 i i)$

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

$4 i (25 - 5 i - 5 i + 1 (i^{1} i))$

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

$4 i (25 - 5 i - 5 i + 1 (i^{1} i^{1}))$

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

$4 i (25 - 5 i - 5 i + 1 i^{1 + 1})$

Add $1$ and $1$ .

$4 i (25 - 5 i - 5 i + 1 i^{2})$

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

$4 i (25 - 5 i - 5 i + i^{2})$

$4 i (25 - 5 i - 5 i + i^{2})$

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

$4 i (25 - 5 i - 5 i - 1)$

$4 i (25 - 5 i - 5 i - 1)$

Subtract $1$ from $25$ .

$4 i (24 - 5 i - 5 i)$

Subtract $5 i$ from $- 5 i$ .

$4 i (24 - 10 i)$

$4 i (24 - 10 i)$

Step 4

Apply the distributive property.

$4 i \cdot 24 + 4 i (- 10 i)$

Step 5

Multiply $24$ by $4$ .

$96 i + 4 i (- 10 i)$

Step 6

Multiply $4 i (- 10 i)$ .

Multiply $- 10$ by $4$ .

$96 i - 40 i i$

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

$96 i - 40 (i^{1} i)$

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

$96 i - 40 (i^{1} i^{1})$

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

$96 i - 40 i^{1 + 1}$

Add $1$ and $1$ .

$96 i - 40 i^{2}$

$96 i - 40 i^{2}$

Step 7

Simplify each term.

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

$96 i - 40 \cdot - 1$

Multiply $- 40$ by $- 1$ .

$96 i + 40$

$96 i + 40$

Step 8

Reorder $96 i$ and $40$ .

$40 + 96 i$

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