Basic Math Examples

$\frac{8 i}{2 - 2 i}$

Step 1

Cancel the common factor of $8$ and $2 - 2 i$ .

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Step 1.1

Factor $2$ out of $8 i$ .

$\frac{2 (4 i)}{2 - 2 i}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $2$ .

$\frac{2 (4 i)}{2 (1) - 2 i}$

Step 1.2.2

Factor $2$ out of $- 2 i$ .

$\frac{2 (4 i)}{2 (1) + 2 (- i)}$

Step 1.2.3

Factor $2$ out of $2 (1) + 2 (- i)$ .

$\frac{2 (4 i)}{2 (1 - i)}$

Step 1.2.4

Cancel the common factor.

$\frac{2 (4 i)}{2 (1 - i)}$

Step 1.2.5

Rewrite the expression.

$\frac{4 i}{1 - i}$

Step 2

Multiply the numerator and denominator of $\frac{4 i}{1 - i}$ by the conjugate of $1 - i$ to make the denominator real.

$\frac{4 i}{1 - i} \cdot \frac{1 + i}{1 + i}$

Step 3

Multiply.

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Step 3.1

Combine.

$\frac{4 i (1 + i)}{(1 - i) (1 + i)}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Apply the distributive property.

$\frac{4 i \cdot 1 + 4 i i}{(1 - i) (1 + i)}$

Step 3.2.2

Multiply $4$ by $1$ .

$\frac{4 i + 4 i i}{(1 - i) (1 + i)}$

Step 3.2.3

Multiply $4 i i$ .

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Step 3.2.3.1

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

$\frac{4 i + 4 (i^{1} i)}{(1 - i) (1 + i)}$

Step 3.2.3.2

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

$\frac{4 i + 4 (i^{1} i^{1})}{(1 - i) (1 + i)}$

Step 3.2.3.3

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

$\frac{4 i + 4 i^{1 + 1}}{(1 - i) (1 + i)}$

Step 3.2.3.4

Add $1$ and $1$ .

$\frac{4 i + 4 i^{2}}{(1 - i) (1 + i)}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

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

$\frac{4 i + 4 \cdot - 1}{(1 - i) (1 + i)}$

Step 3.2.4.2

Multiply $4$ by $- 1$ .

$\frac{4 i - 4}{(1 - i) (1 + i)}$

Step 3.2.5

Reorder $4 i$ and $- 4$ .

$\frac{- 4 + 4 i}{(1 - i) (1 + i)}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Expand $(1 - i) (1 + i)$ using the FOIL Method.

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Step 3.3.1.1

Apply the distributive property.

$\frac{- 4 + 4 i}{1 (1 + i) - i (1 + i)}$

Step 3.3.1.2

Apply the distributive property.

$\frac{- 4 + 4 i}{1 \cdot 1 + 1 i - i (1 + i)}$

Step 3.3.1.3

Apply the distributive property.

$\frac{- 4 + 4 i}{1 \cdot 1 + 1 i - i \cdot 1 - i i}$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Multiply $1$ by $1$ .

$\frac{- 4 + 4 i}{1 + 1 i - i \cdot 1 - i i}$

Step 3.3.2.2

Multiply $- 1$ by $1$ .

$\frac{- 4 + 4 i}{1 + 1 i - i - i i}$

Step 3.3.2.3

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

$\frac{- 4 + 4 i}{1 + 1 i - i - (i^{1} i)}$

Step 3.3.2.4

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

$\frac{- 4 + 4 i}{1 + 1 i - i - (i^{1} i^{1})}$

Step 3.3.2.5

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

$\frac{- 4 + 4 i}{1 + 1 i - i - i^{1 + 1}}$

Step 3.3.2.6

Add $1$ and $1$ .

$\frac{- 4 + 4 i}{1 + 1 i - i - i^{2}}$

Step 3.3.2.7

Subtract $i$ from $1 i$ .

$\frac{- 4 + 4 i}{1 + 0 - i^{2}}$

Step 3.3.2.8

Add $1$ and $0$ .

$\frac{- 4 + 4 i}{1 - i^{2}}$

Step 3.3.3

Simplify each term.

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Step 3.3.3.1

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

$\frac{- 4 + 4 i}{1 - - 1}$

Step 3.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{- 4 + 4 i}{1 + 1}$

Step 3.3.4

Add $1$ and $1$ .

$\frac{- 4 + 4 i}{2}$

Step 4

Cancel the common factor of $- 4 + 4 i$ and $2$ .

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Step 4.1

Factor $2$ out of $- 4$ .

$\frac{2 \cdot - 2 + 4 i}{2}$

Step 4.2

Factor $2$ out of $4 i$ .

$\frac{2 \cdot - 2 + 2 (2 i)}{2}$

Step 4.3

Factor $2$ out of $2 (- 2) + 2 (2 i)$ .

$\frac{2 (- 2 + 2 i)}{2}$

Step 4.4

Cancel the common factors.

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Step 4.4.1

Factor $2$ out of $2$ .

$\frac{2 (- 2 + 2 i)}{2 (1)}$

Step 4.4.2

Cancel the common factor.

$\frac{2 (- 2 + 2 i)}{2 \cdot 1}$

Step 4.4.3

Rewrite the expression.

$\frac{- 2 + 2 i}{1}$

Step 4.4.4

Divide $- 2 + 2 i$ by $1$ .

$- 2 + 2 i$