Finite Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(1 + i)}^{2}$ as $(1 + i) (1 + i)$ .

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $1$ by $1$ .

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

Step 1.3.1.2

Multiply $i$ by $1$ .

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

Step 1.3.1.3

Multiply $i$ by $1$ .

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

Step 1.3.1.4

Multiply $i i$ .

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Step 1.3.1.4.1

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

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

Step 1.3.1.4.2

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

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

Step 1.3.1.4.3

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

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

Step 1.3.1.4.4

Add $1$ and $1$ .

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

Step 1.3.1.5

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

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

Step 1.3.2

Subtract $1$ from $1$ .

$\frac{0 + i + i}{{(1 - i)}^{2}} + \frac{2}{x + i y}$

Step 1.3.3

Add $0$ and $i$ .

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

Step 1.3.4

Add $i$ and $i$ .

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

Step 1.4

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

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

Step 1.5

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

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

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

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $1$ by $1$ .

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

Step 1.6.1.2

Multiply $- i$ by $1$ .

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

Step 1.6.1.3

Multiply $- 1$ by $1$ .

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

Step 1.6.1.4

Multiply $- i (- i)$ .

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Step 1.6.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.6.1.4.2

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

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

Step 1.6.1.4.3

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

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

Step 1.6.1.4.4

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

$\frac{2 i}{1 - i - i + 1 i^{1 + 1}} + \frac{2}{x + i y}$

Step 1.6.1.4.5

Add $1$ and $1$ .

$\frac{2 i}{1 - i - i + 1 i^{2}} + \frac{2}{x + i y}$

Step 1.6.1.4.6

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

$\frac{2 i}{1 - i - i + i^{2}} + \frac{2}{x + i y}$

Step 1.6.1.5

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

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

Step 1.6.2

Subtract $1$ from $1$ .

$\frac{2 i}{0 - i - i} + \frac{2}{x + i y}$

Step 1.6.3

Subtract $i$ from $0$ .

$\frac{2 i}{- i - i} + \frac{2}{x + i y}$

Step 1.6.4

Subtract $i$ from $- i$ .

$\frac{2 i}{- 2 i} + \frac{2}{x + i y}$

Step 1.7

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

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Step 1.7.1

Factor $2$ out of $2 i$ .

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

Step 1.7.2

Cancel the common factors.

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Step 1.7.2.1

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

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

Step 1.7.2.2

Cancel the common factor.

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

Step 1.7.2.3

Rewrite the expression.

$\frac{i}{- i} + \frac{2}{x + i y}$

Step 1.8

Cancel the common factor of $i$ .

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Step 1.8.1

Cancel the common factor.

$\frac{i}{- i} + \frac{2}{x + i y}$

Step 1.8.2

Rewrite the expression.

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

Step 1.8.3

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$- 1 \cdot 1 + \frac{2}{x + i y}$

Step 1.9

Multiply $- 1$ by $1$ .

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

Step 2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x + i y}{x + i y}$ .

$- 1 \cdot \frac{x + i y}{x + i y} + \frac{2}{x + i y}$

Step 3

Simplify terms.

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Step 3.1

Combine $- 1$ and $\frac{x + i y}{x + i y}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Apply the distributive property.

$\frac{- x - i y + 2}{x + i y}$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $- 1$ out of $- x$ .

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

Step 5.2

Factor $- 1$ out of $- i y$ .

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

Step 5.3

Factor $- 1$ out of $- (x) - (i y)$ .

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

Step 5.4

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 5.5

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

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

Step 5.6

Simplify the expression.

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Step 5.6.1

Rewrite $- (x + i y - 2)$ as $- 1 (x + i y - 2)$ .

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

Step 5.6.2

Move the negative in front of the fraction.

$- \frac{x + i y - 2}{x + i y}$

Step 6

Split the fraction $\frac{x + i y - 2}{x + i y}$ into two fractions.

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

Step 7

Split the fraction $\frac{x + i y}{x + i y}$ into two fractions.

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

Step 8

Move the negative in front of the fraction.

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