Basic Math Examples

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

Step 1

Cancel the common factor of

$4 + 4 i$ and

$2 - 2 i$ .

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Step 1.1

Factor

$2$ out of

$4$ .

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

Step 1.2

Factor

$2$ out of

$4 i$ .

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

Step 1.3

Factor

$2$ out of

$2 (2) + 2 (2 i)$ .

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

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor

$2$ out of

$2$ .

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

Step 1.4.2

Factor

$2$ out of

$- 2 i$ .

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

Step 1.4.3

Factor

$2$ out of

$2 (1) + 2 (- i)$ .

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

Step 1.4.4

Cancel the common factor.

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

Step 1.4.5

Rewrite the expression.

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

Step 2

Multiply the numerator and denominator of

$\frac{2 + 2 i}{1 - i}$ by the conjugate of

$1 - i$ to make the denominator real.

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

Step 3

Multiply.

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Step 3.1

Combine.

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

Expand

$(2 + 2 i) (1 + i)$ using the FOIL Method.

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Step 3.2.1.1

Apply the distributive property.

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

Step 3.2.1.2

Apply the distributive property.

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

Step 3.2.1.3

Apply the distributive property.

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

Step 3.2.2

Simplify and combine like terms.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply

$2$ by

$1$ .

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

Step 3.2.2.1.2

Multiply

$2$ by

$1$ .

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

Step 3.2.2.1.3

Multiply

$2 i i$ .

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Step 3.2.2.1.3.1

Raise

$i$ to the power of

$1$ .

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

Step 3.2.2.1.3.2

Raise

$i$ to the power of

$1$ .

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

Step 3.2.2.1.3.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2.2.1.3.4

Add

$1$ and

$1$ .

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

Step 3.2.2.1.4

Rewrite

$i^{2}$ as

$- 1$ .

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

Step 3.2.2.1.5

Multiply

$2$ by

$- 1$ .

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

Step 3.2.2.2

Subtract

$2$ from

$2$ .

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

Step 3.2.2.3

Add

$0$ and

$2 i$ .

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

Step 3.2.2.4

Add

$2 i$ and

$2 i$ .

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

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.

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

Step 3.3.1.2

Apply the distributive property.

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

Step 3.3.1.3

Apply the distributive property.

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

Step 3.3.2

Simplify.

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Step 3.3.2.1

Multiply

$1$ by

$1$ .

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

Step 3.3.2.2

Multiply

$- 1$ by

$1$ .

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

Step 3.3.2.3

Raise

$i$ to the power of

$1$ .

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

Step 3.3.2.4

Raise

$i$ to the power of

$1$ .

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

Step 3.3.2.5

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.3.2.6

Add

$1$ and

$1$ .

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

Step 3.3.2.7

Subtract

$i$ from

$1 i$ .

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

Step 3.3.2.8

Add

$1$ and

$0$ .

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

Step 3.3.3

Simplify each term.

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Step 3.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

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

Step 3.3.3.2

Multiply

$- 1$ by

$- 1$ .

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

Step 3.3.4

Add

$1$ and

$1$ .

$\frac{4 i}{2}$

Step 4

Cancel the common factor of

$4$ and

$2$ .

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Step 4.1

Factor

$2$ out of

$4 i$ .

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor

$2$ out of

$2$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

$\frac{2 i}{1}$

Step 4.2.4

Divide

$2 i$ by

$1$ .

$2 i$