Finite Math Examples

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

Step 1

To write $\frac{{(1 + i)}^{2}}{{(1 - i)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x - i y}{x - i y}$ .

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

Step 2

To write $\frac{2}{x - i y}$ as a fraction with a common denominator, multiply by $\frac{{(1 - i)}^{2}}{{(1 - i)}^{2}}$ .

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

Step 3

Write each expression with a common denominator of $(x - i y) {(1 - i)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

Multiply $\frac{2}{x - i y}$ by $\frac{{(1 - i)}^{2}}{{(1 - i)}^{2}}$ .

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

Step 3.3

Reorder the factors of $(x - i y) {(1 - i)}^{2}$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Rewrite $\frac{{(1 + i)}^{2} (x - i y) + 2 {(1 - i)}^{2}}{{(1 - i)}^{2} (x - i y)}$ in a factored form.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

Apply the distributive property.

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

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $1$ by $1$ .

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

Step 5.3.1.2

Multiply $i$ by $1$ .

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

Step 5.3.1.3

Multiply $i$ by $1$ .

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

Step 5.3.1.4

Multiply $i i$ .

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Step 5.3.1.4.1

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

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

Step 5.3.1.4.2

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

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

Step 5.3.1.4.3

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

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

Step 5.3.1.4.4

Add $1$ and $1$ .

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

Step 5.3.1.5

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

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

Step 5.3.2

Subtract $1$ from $1$ .

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

Step 5.3.3

Add $0$ and $i$ .

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

Step 5.3.4

Add $i$ and $i$ .

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

Step 5.4

Apply the distributive property.

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

Step 5.5

Multiply $2 i (- i y)$ .

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Step 5.5.1

Multiply $- 1$ by $2$ .

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

Step 5.5.2

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

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

Step 5.5.3

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

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

Step 5.5.4

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

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

Step 5.5.5

Add $1$ and $1$ .

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

Step 5.6

Simplify each term.

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Step 5.6.1

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

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

Step 5.6.2

Multiply $- 2$ by $- 1$ .

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

Step 5.7

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

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

Step 5.8

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

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Step 5.8.1

Apply the distributive property.

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

Step 5.8.2

Apply the distributive property.

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

Step 5.8.3

Apply the distributive property.

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

Step 5.9

Simplify and combine like terms.

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Step 5.9.1

Simplify each term.

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Step 5.9.1.1

Multiply $1$ by $1$ .

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

Step 5.9.1.2

Multiply $- i$ by $1$ .

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

Step 5.9.1.3

Multiply $- 1$ by $1$ .

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

Step 5.9.1.4

Multiply $- i (- i)$ .

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Step 5.9.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 5.9.1.4.2

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

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

Step 5.9.1.4.3

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

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

Step 5.9.1.4.4

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

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

Step 5.9.1.4.5

Add $1$ and $1$ .

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

Step 5.9.1.4.6

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

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

Step 5.9.1.5

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

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

Step 5.9.2

Subtract $1$ from $1$ .

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

Step 5.9.3

Subtract $i$ from $0$ .

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

Step 5.9.4

Subtract $i$ from $- i$ .

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

Step 5.10

Multiply $- 2$ by $2$ .

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

Step 5.11

Factor $2$ out of $2 i x + 2 y - 4 i$ .

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Step 5.11.1

Factor $2$ out of $2 i x$ .

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

Step 5.11.2

Factor $2$ out of $2 y$ .

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

Step 5.11.3

Factor $2$ out of $- 4 i$ .

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

Step 5.11.4

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

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

Step 5.11.5

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

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