Finite Math Examples

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

Step 1

Multiply the numerator and denominator of $\frac{{(2 i)}^{2}}{4 - 2 i}$ by the conjugate of $4 - 2 i$ to make the denominator real.

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the product rule to $2 i$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $4$ by $- 1$ .

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

Step 2.2.5

Apply the distributive property.

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

Step 2.2.6

Multiply $- 4$ by $4$ .

$\frac{- 16 - 4 (2 i)}{(4 - 2 i) (4 + 2 i)}$

Step 2.2.7

Multiply $2$ by $- 4$ .

$\frac{- 16 - 8 i}{(4 - 2 i) (4 + 2 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 16 - 8 i}{4 (4 + 2 i) - 2 i (4 + 2 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 16 - 8 i}{4 \cdot 4 + 4 (2 i) - 2 i (4 + 2 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 16 - 8 i}{4 \cdot 4 + 4 (2 i) - 2 i \cdot 4 - 2 i (2 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $4$ by $4$ .

$\frac{- 16 - 8 i}{16 + 4 (2 i) - 2 i \cdot 4 - 2 i (2 i)}$

Step 2.3.2.2

Multiply $2$ by $4$ .

$\frac{- 16 - 8 i}{16 + 8 i - 2 i \cdot 4 - 2 i (2 i)}$

Step 2.3.2.3

Multiply $4$ by $- 2$ .

$\frac{- 16 - 8 i}{16 + 8 i - 8 i - 2 i (2 i)}$

Step 2.3.2.4

Multiply $2$ by $- 2$ .

$\frac{- 16 - 8 i}{16 + 8 i - 8 i - 4 i i}$

Step 2.3.2.5

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

$\frac{- 16 - 8 i}{16 + 8 i - 8 i - 4 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{- 16 - 8 i}{16 + 8 i - 8 i - 4 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{- 16 - 8 i}{16 + 8 i - 8 i - 4 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{- 16 - 8 i}{16 + 8 i - 8 i - 4 i^{2}}$

Step 2.3.2.9

Subtract $8 i$ from $8 i$ .

$\frac{- 16 - 8 i}{16 + 0 - 4 i^{2}}$

Step 2.3.2.10

Add $16$ and $0$ .

$\frac{- 16 - 8 i}{16 - 4 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{- 16 - 8 i}{16 - 4 \cdot - 1}$

Step 2.3.3.2

Multiply $- 4$ by $- 1$ .

$\frac{- 16 - 8 i}{16 + 4}$

Step 2.3.4

Add $16$ and $4$ .

$\frac{- 16 - 8 i}{20}$

Step 3

Cancel the common factor of $- 16 - 8 i$ and $20$ .

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Step 3.1

Factor $4$ out of $- 16$ .

$\frac{4 (- 4) - 8 i}{20}$

Step 3.2

Factor $4$ out of $- 8 i$ .

$\frac{4 (- 4) + 4 (- 2 i)}{20}$

Step 3.3

Factor $4$ out of $4 (- 4) + 4 (- 2 i)$ .

$\frac{4 (- 4 - 2 i)}{20}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $4$ out of $20$ .

$\frac{4 (- 4 - 2 i)}{4 \cdot 5}$

Step 3.4.2

Cancel the common factor.

$\frac{4 (- 4 - 2 i)}{4 \cdot 5}$

Step 3.4.3

Rewrite the expression.

$\frac{- 4 - 2 i}{5}$

Step 4

Split the fraction $\frac{- 4 - 2 i}{5}$ into two fractions.

$\frac{- 4}{5} + \frac{- 2 i}{5}$

Step 5

Simplify each term.

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Step 5.1

Move the negative in front of the fraction.

$- \frac{4}{5} + \frac{- 2 i}{5}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{4}{5} - \frac{2 i}{5}$