Algebra Examples

\frac{2 i}{1 + i}

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

Step 1

Multiply the numerator and denominator of

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

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

1 + 1 i

$1 + 1 i$ to make the denominator real.

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

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

Step 2

Multiply.

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Step 2.1

Combine.

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

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

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

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

Step 2.2.2

Multiply

2

$2$ by

1

$1$ .

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

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

Step 2.2.3

Multiply

2 i (- i)

$2 i (- i)$ .

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Step 2.2.3.1

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

Step 2.2.3.2

Raise

i

$i$ to the power of

1

$1$ .

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

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

Step 2.2.3.3

Raise

i

$i$ to the power of

1

$1$ .

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

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

Step 2.2.3.4

Use the power rule

a^{m} a^{n} = a^{m + n}

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

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

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

Step 2.2.3.5

Add

1

$1$ and

1

$1$ .

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

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

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

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

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

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

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

Step 2.2.4.2

Multiply

- 2

$- 2$ by

- 1

$- 1$ .

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

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

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

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

Step 2.2.5

Reorder

2 i

$2 i$ and

2

$2$ .

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

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

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

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

(1 + 1 i) (1 - i)

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

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Step 2.3.1.1

Apply the distributive property.

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

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

Step 2.3.1.2

Apply the distributive property.

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

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

Step 2.3.1.3

Apply the distributive property.

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

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

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

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

1

$1$ by

1

$1$ .

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

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

Step 2.3.2.2

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

Step 2.3.2.3

Multiply

1

$1$ by

1

$1$ .

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

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

Step 2.3.2.4

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

Step 2.3.2.5

Raise

i

$i$ to the power of

1

$1$ .

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

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

Step 2.3.2.6

Raise

i

$i$ to the power of

1

$1$ .

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

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

Step 2.3.2.7

Use the power rule

a^{m} a^{n} = a^{m + n}

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

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

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

Step 2.3.2.8

Add

1

$1$ and

1

$1$ .

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

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

Step 2.3.2.9

Add

- 1 i

$- 1 i$ and

1 i

$1 i$ .

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

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

Step 2.3.2.10

Add

1

$1$ and

0

$0$ .

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

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

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

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

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

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

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

Step 2.3.3.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

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

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

Step 2.3.4

Add

1

$1$ and

1

$1$ .

\frac{2 + 2 i}{2}

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

\frac{2 + 2 i}{2}

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

\frac{2 + 2 i}{2}

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

Step 3

Cancel the common factor of

2 + 2 i

$2 + 2 i$ and

2

$2$ .

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Step 3.1

Factor

2

$2$ out of

2

$2$ .

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

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

Step 3.2

Factor

2

$2$ out of

2 i

$2 i$ .

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

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

Step 3.3

Factor

2

$2$ out of

2 (1) + 2 (i)

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

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

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

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor

2

$2$ out of

2

$2$ .

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

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 3.4.4

Divide

$1 + i$ by

$1$ .

$1 + i$