Simplify i^5-4+(2-4i)(-3+5i)

$i^{5} - 4 + (2 - 4 i) (- 3 + 5 i)$

Step 1

Simplify each term.

Factor out $i^{4}$ .

$i^{4} i - 4 + (2 - 4 i) (- 3 + 5 i)$

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

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

${(i^{2})}^{2} i - 4 + (2 - 4 i) (- 3 + 5 i)$

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

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

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

$1 i - 4 + (2 - 4 i) (- 3 + 5 i)$

Multiply $i$ by $1$ .

$i - 4 + (2 - 4 i) (- 3 + 5 i)$

Expand $(2 - 4 i) (- 3 + 5 i)$ using the FOIL Method.

Apply the distributive property.

$i - 4 + 2 (- 3 + 5 i) - 4 i (- 3 + 5 i)$

Apply the distributive property.

$i - 4 + 2 \cdot - 3 + 2 (5 i) - 4 i (- 3 + 5 i)$

Apply the distributive property.

$i - 4 + 2 \cdot - 3 + 2 (5 i) - 4 i \cdot - 3 - 4 i (5 i)$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $- 3$ .

$i - 4 - 6 + 2 (5 i) - 4 i \cdot - 3 - 4 i (5 i)$

Multiply $5$ by $2$ .

$i - 4 - 6 + 10 i - 4 i \cdot - 3 - 4 i (5 i)$

Multiply $- 3$ by $- 4$ .

$i - 4 - 6 + 10 i + 12 i - 4 i (5 i)$

Multiply $- 4 i (5 i)$ .

Multiply $5$ by $- 4$ .

$i - 4 - 6 + 10 i + 12 i - 20 i i$

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

$i - 4 - 6 + 10 i + 12 i - 20 (i^{1} i)$

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

$i - 4 - 6 + 10 i + 12 i - 20 (i^{1} i^{1})$

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

$i - 4 - 6 + 10 i + 12 i - 20 i^{1 + 1}$

Add $1$ and $1$ .

$i - 4 - 6 + 10 i + 12 i - 20 i^{2}$

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

$i - 4 - 6 + 10 i + 12 i - 20 \cdot - 1$

Multiply $- 20$ by $- 1$ .

$i - 4 - 6 + 10 i + 12 i + 20$

Add $- 6$ and $20$ .

$i - 4 + 14 + 10 i + 12 i$

Add $10 i$ and $12 i$ .

$i - 4 + 14 + 22 i$

Step 2

Simplify by adding terms.

Add $i$ and $22 i$ .

$- 4 + 14 + 23 i$

Add $- 4$ and $14$ .

$10 + 23 i$

Simplify i^5-4+(2-4i)(-3+5i)

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