Simplify (5-i^3)-(8+i^7)

$(5 - i^{3}) - (8 + i^{7})$

Simplify each term.

Factor out $i^{2}$ .

$5 - (i^{2} \cdot i) - (8 + i^{7})$

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

$5 - (- 1 \cdot i) - (8 + i^{7})$

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

$5 - - i - (8 + i^{7})$

Multiply $- 1$ by $- 1$ .

$5 + 1 i - (8 + i^{7})$

Multiply $i$ by $1$ .

$5 + i - (8 + i^{7})$

Simplify each term.

Rewrite $i^{7}$ as $i^{4} (i^{2} \cdot i)$ .

Factor out $i^{4}$ .

$5 + i - (8 + i^{4} i^{3})$

Factor out $i^{2}$ .

$5 + i - (8 + i^{4} (i^{2} \cdot i))$

$5 + i - (8 + i^{4} (i^{2} \cdot i))$

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

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

$5 + i - (8 + {(i^{2})}^{2} (i^{2} \cdot i))$

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

$5 + i - (8 + {(- 1)}^{2} (i^{2} \cdot i))$

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

$5 + i - (8 + 1 (i^{2} \cdot i))$

$5 + i - (8 + 1 (i^{2} \cdot i))$

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

$5 + i - (8 + i^{2} \cdot i)$

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

$5 + i - (8 - 1 \cdot i)$

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

$5 + i - (8 - i)$

$5 + i - (8 - i)$

Apply the distributive property.

$5 + i - 1 \cdot 8 - - i$

Multiply $- 1$ by $8$ .

$5 + i - 8 - - i$

Multiply $- 1$ by $- 1$ .

$5 + i - 8 + 1 i$

Multiply $i$ by $1$ .

$5 + i - 8 + i$

$5 + i - 8 + i$

Simplify by adding terms.

Subtract $8$ from $5$ .

$- 3 + i + i$

Add $i$ and $i$ .

$- 3 + 2 i$

$- 3 + 2 i$

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