Basic Math Examples

\frac{2 - i}{3 + 2 i}

$\frac{2 - i}{3 + 2 i}$

Step 1

Multiply the numerator and denominator of

\frac{2 - i}{3 + 2 i}

$\frac{2 - i}{3 + 2 i}$ by the conjugate of

3 + 2 i

$3 + 2 i$ to make the denominator real.

\frac{2 - i}{3 + 2 i} \cdot \frac{3 - 2 i}{3 - 2 i}

$\frac{2 - i}{3 + 2 i} \cdot \frac{3 - 2 i}{3 - 2 i}$

Step 2

Multiply.

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Step 2.1

Combine.

\frac{(2 - i) (3 - 2 i)}{(3 + 2 i) (3 - 2 i)}

$\frac{(2 - i) (3 - 2 i)}{(3 + 2 i) (3 - 2 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(2 - i) (3 - 2 i)

$(2 - i) (3 - 2 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

\frac{2 (3 - 2 i) - i (3 - 2 i)}{(3 + 2 i) (3 - 2 i)}

$\frac{2 (3 - 2 i) - i (3 - 2 i)}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.1.2

Apply the distributive property.

\frac{2 \cdot 3 + 2 (- 2 i) - i (3 - 2 i)}{(3 + 2 i) (3 - 2 i)}

$\frac{2 \cdot 3 + 2 (- 2 i) - i (3 - 2 i)}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.1.3

Apply the distributive property.

\frac{2 \cdot 3 + 2 (- 2 i) - i \cdot 3 - i (- 2 i)}{(3 + 2 i) (3 - 2 i)}

$\frac{2 \cdot 3 + 2 (- 2 i) - i \cdot 3 - i (- 2 i)}{(3 + 2 i) (3 - 2 i)}$

\frac{2 \cdot 3 + 2 (- 2 i) - i \cdot 3 - i (- 2 i)}{(3 + 2 i) (3 - 2 i)}

$\frac{2 \cdot 3 + 2 (- 2 i) - i \cdot 3 - i (- 2 i)}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.2

Simplify and combine like terms.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply

2

$2$ by

3

$3$ .

\frac{6 + 2 (- 2 i) - i \cdot 3 - i (- 2 i)}{(3 + 2 i) (3 - 2 i)}

$\frac{6 + 2 (- 2 i) - i \cdot 3 - i (- 2 i)}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.2.1.2

Multiply

- 2

$- 2$ by

2

$2$ .

\frac{6 - 4 i - i \cdot 3 - i (- 2 i)}{(3 + 2 i) (3 - 2 i)}

$\frac{6 - 4 i - i \cdot 3 - i (- 2 i)}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.2.1.3

Multiply

3

$3$ by

- 1

$- 1$ .

\frac{6 - 4 i - 3 i - i (- 2 i)}{(3 + 2 i) (3 - 2 i)}

$\frac{6 - 4 i - 3 i - i (- 2 i)}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.2.1.4

Multiply

- i (- 2 i)

$- i (- 2 i)$ .

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Step 2.2.2.1.4.1

Multiply

- 2

$- 2$ by

- 1

$- 1$ .

\frac{6 - 4 i - 3 i + 2 i i}{(3 + 2 i) (3 - 2 i)}

$\frac{6 - 4 i - 3 i + 2 i i}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.2.1.4.2

Raise

i

$i$ to the power of

1

$1$ .

\frac{6 - 4 i - 3 i + 2 (i^{1} i)}{(3 + 2 i) (3 - 2 i)}

$\frac{6 - 4 i - 3 i + 2 (i^{1} i)}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.2.1.4.3

Raise

i

$i$ to the power of

1

$1$ .

\frac{6 - 4 i - 3 i + 2 (i^{1} i^{1})}{(3 + 2 i) (3 - 2 i)}

$\frac{6 - 4 i - 3 i + 2 (i^{1} i^{1})}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.2.1.4.4

Use the power rule

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

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

\frac{6 - 4 i - 3 i + 2 i^{1 + 1}}{(3 + 2 i) (3 - 2 i)}

$\frac{6 - 4 i - 3 i + 2 i^{1 + 1}}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.2.1.4.5

Add

1

$1$ and

1

$1$ .

\frac{6 - 4 i - 3 i + 2 i^{2}}{(3 + 2 i) (3 - 2 i)}

$\frac{6 - 4 i - 3 i + 2 i^{2}}{(3 + 2 i) (3 - 2 i)}$

\frac{6 - 4 i - 3 i + 2 i^{2}}{(3 + 2 i) (3 - 2 i)}

$\frac{6 - 4 i - 3 i + 2 i^{2}}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.2.1.5

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{6 - 4 i - 3 i + 2 \cdot - 1}{(3 + 2 i) (3 - 2 i)}

$\frac{6 - 4 i - 3 i + 2 \cdot - 1}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.2.1.6

Multiply

2

$2$ by

- 1

$- 1$ .

\frac{6 - 4 i - 3 i - 2}{(3 + 2 i) (3 - 2 i)}

$\frac{6 - 4 i - 3 i - 2}{(3 + 2 i) (3 - 2 i)}$

\frac{6 - 4 i - 3 i - 2}{(3 + 2 i) (3 - 2 i)}

$\frac{6 - 4 i - 3 i - 2}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.2.2

Subtract

2

$2$ from

6

$6$ .

\frac{4 - 4 i - 3 i}{(3 + 2 i) (3 - 2 i)}

$\frac{4 - 4 i - 3 i}{(3 + 2 i) (3 - 2 i)}$

Step 2.2.2.3

Subtract

3 i

$3 i$ from

- 4 i

$- 4 i$ .

\frac{4 - 7 i}{(3 + 2 i) (3 - 2 i)}

$\frac{4 - 7 i}{(3 + 2 i) (3 - 2 i)}$

\frac{4 - 7 i}{(3 + 2 i) (3 - 2 i)}

$\frac{4 - 7 i}{(3 + 2 i) (3 - 2 i)}$

\frac{4 - 7 i}{(3 + 2 i) (3 - 2 i)}

$\frac{4 - 7 i}{(3 + 2 i) (3 - 2 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

(3 + 2 i) (3 - 2 i)

$(3 + 2 i) (3 - 2 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

\frac{4 - 7 i}{3 (3 - 2 i) + 2 i (3 - 2 i)}

$\frac{4 - 7 i}{3 (3 - 2 i) + 2 i (3 - 2 i)}$

Step 2.3.1.2

Apply the distributive property.

\frac{4 - 7 i}{3 \cdot 3 + 3 (- 2 i) + 2 i (3 - 2 i)}

$\frac{4 - 7 i}{3 \cdot 3 + 3 (- 2 i) + 2 i (3 - 2 i)}$

Step 2.3.1.3

Apply the distributive property.

\frac{4 - 7 i}{3 \cdot 3 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}

$\frac{4 - 7 i}{3 \cdot 3 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}$

\frac{4 - 7 i}{3 \cdot 3 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}

$\frac{4 - 7 i}{3 \cdot 3 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

3

$3$ by

3

$3$ .

\frac{4 - 7 i}{9 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}

$\frac{4 - 7 i}{9 + 3 (- 2 i) + 2 i \cdot 3 + 2 i (- 2 i)}$

Step 2.3.2.2

Multiply

- 2

$- 2$ by

3

$3$ .

\frac{4 - 7 i}{9 - 6 i + 2 i \cdot 3 + 2 i (- 2 i)}

$\frac{4 - 7 i}{9 - 6 i + 2 i \cdot 3 + 2 i (- 2 i)}$

Step 2.3.2.3

Multiply

3

$3$ by

2

$2$ .

\frac{4 - 7 i}{9 - 6 i + 6 i + 2 i (- 2 i)}

$\frac{4 - 7 i}{9 - 6 i + 6 i + 2 i (- 2 i)}$

Step 2.3.2.4

Multiply

- 2

$- 2$ by

2

$2$ .

\frac{4 - 7 i}{9 - 6 i + 6 i - 4 i i}

$\frac{4 - 7 i}{9 - 6 i + 6 i - 4 i i}$

Step 2.3.2.5

Raise

i

$i$ to the power of

1

$1$ .

\frac{4 - 7 i}{9 - 6 i + 6 i - 4 (i^{1} i)}

$\frac{4 - 7 i}{9 - 6 i + 6 i - 4 (i^{1} i)}$

Step 2.3.2.6

Raise

i

$i$ to the power of

1

$1$ .

\frac{4 - 7 i}{9 - 6 i + 6 i - 4 (i^{1} i^{1})}

$\frac{4 - 7 i}{9 - 6 i + 6 i - 4 (i^{1} i^{1})}$

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{4 - 7 i}{9 - 6 i + 6 i - 4 i^{1 + 1}}

$\frac{4 - 7 i}{9 - 6 i + 6 i - 4 i^{1 + 1}}$

Step 2.3.2.8

Add

1

$1$ and

1

$1$ .

\frac{4 - 7 i}{9 - 6 i + 6 i - 4 i^{2}}

$\frac{4 - 7 i}{9 - 6 i + 6 i - 4 i^{2}}$

Step 2.3.2.9

Add

- 6 i

$- 6 i$ and

6 i

$6 i$ .

\frac{4 - 7 i}{9 + 0 - 4 i^{2}}

$\frac{4 - 7 i}{9 + 0 - 4 i^{2}}$

Step 2.3.2.10

Add

9

$9$ and

0

$0$ .

\frac{4 - 7 i}{9 - 4 i^{2}}

$\frac{4 - 7 i}{9 - 4 i^{2}}$

\frac{4 - 7 i}{9 - 4 i^{2}}

$\frac{4 - 7 i}{9 - 4 i^{2}}$

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{4 - 7 i}{9 - 4 \cdot - 1}

$\frac{4 - 7 i}{9 - 4 \cdot - 1}$

Step 2.3.3.2

Multiply

- 4

$- 4$ by

- 1

$- 1$ .

\frac{4 - 7 i}{9 + 4}

$\frac{4 - 7 i}{9 + 4}$

\frac{4 - 7 i}{9 + 4}

$\frac{4 - 7 i}{9 + 4}$

Step 2.3.4

Add

9

$9$ and

4

$4$ .

\frac{4 - 7 i}{13}

$\frac{4 - 7 i}{13}$

\frac{4 - 7 i}{13}

$\frac{4 - 7 i}{13}$

\frac{4 - 7 i}{13}

$\frac{4 - 7 i}{13}$

Step 3

Split the fraction

\frac{4 - 7 i}{13}

$\frac{4 - 7 i}{13}$ into two fractions.

\frac{4}{13} + \frac{- 7 i}{13}

$\frac{4}{13} + \frac{- 7 i}{13}$

Step 4

Move the negative in front of the fraction.

\frac{4}{13} - \frac{7 i}{13}

$\frac{4}{13} - \frac{7 i}{13}$