Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Factor out $i^{4}$ .

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

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

Step 1.3

Multiply $i$ by $1$ .

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

Step 1.4

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

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Step 1.4.1

Apply the distributive property.

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

Step 1.4.2

Apply the distributive property.

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

Step 1.4.3

Apply the distributive property.

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

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

Step 1.5

Simplify and combine like terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Multiply $2$ by $- 3$ .

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

Step 1.5.1.2

Multiply $5$ by $2$ .

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

Step 1.5.1.3

Multiply $- 3$ by $- 4$ .

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

Step 1.5.1.4

Multiply $- 4 i (5 i)$ .

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Step 1.5.1.4.1

Multiply $5$ by $- 4$ .

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

Step 1.5.1.4.2

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

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

Step 1.5.1.4.3

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

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

Step 1.5.1.4.4

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

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

Step 1.5.1.4.5

Add $1$ and $1$ .

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

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

Step 1.5.1.5

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

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

Step 1.5.1.6

Multiply $- 20$ by $- 1$ .

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

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

Step 1.5.2

Add $- 6$ and $20$ .

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

Step 1.5.3

Add $10 i$ and $12 i$ .

$i - 4 + 14 + 22 i$

$i - 4 + 14 + 22 i$

$i - 4 + 14 + 22 i$

Step 2

Simplify by adding terms.

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Step 2.1

Add $i$ and $22 i$ .

$- 4 + 14 + 23 i$

Step 2.2

Add $- 4$ and $14$ .

$10 + 23 i$

$10 + 23 i$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$