Basic Math Examples

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

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

Step 1

Multiply the numerator and denominator of

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

$\frac{2 - 3 i}{6 + 4 i}$ by the conjugate of

6 + 4 i

$6 + 4 i$ to make the denominator real.

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

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

Step 2

Multiply.

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Step 2.1

Combine.

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

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(2 - 3 i) (6 - 4 i)

$(2 - 3 i) (6 - 4 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

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

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

Step 2.2.1.2

Apply the distributive property.

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

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

Step 2.2.1.3

Apply the distributive property.

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

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

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

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

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

6

$6$ .

\frac{12 + 2 (- 4 i) - 3 i \cdot 6 - 3 i (- 4 i)}{(6 + 4 i) (6 - 4 i)}

$\frac{12 + 2 (- 4 i) - 3 i \cdot 6 - 3 i (- 4 i)}{(6 + 4 i) (6 - 4 i)}$

Step 2.2.2.1.2

Multiply

- 4

$- 4$ by

2

$2$ .

\frac{12 - 8 i - 3 i \cdot 6 - 3 i (- 4 i)}{(6 + 4 i) (6 - 4 i)}

$\frac{12 - 8 i - 3 i \cdot 6 - 3 i (- 4 i)}{(6 + 4 i) (6 - 4 i)}$

Step 2.2.2.1.3

Multiply

6

$6$ by

- 3

$- 3$ .

\frac{12 - 8 i - 18 i - 3 i (- 4 i)}{(6 + 4 i) (6 - 4 i)}

$\frac{12 - 8 i - 18 i - 3 i (- 4 i)}{(6 + 4 i) (6 - 4 i)}$

Step 2.2.2.1.4

Multiply

- 3 i (- 4 i)

$- 3 i (- 4 i)$ .

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Step 2.2.2.1.4.1

Multiply

- 4

$- 4$ by

- 3

$- 3$ .

\frac{12 - 8 i - 18 i + 12 i i}{(6 + 4 i) (6 - 4 i)}

$\frac{12 - 8 i - 18 i + 12 i i}{(6 + 4 i) (6 - 4 i)}$

Step 2.2.2.1.4.2

Raise

i

$i$ to the power of

1

$1$ .

\frac{12 - 8 i - 18 i + 12 (i^{1} i)}{(6 + 4 i) (6 - 4 i)}

$\frac{12 - 8 i - 18 i + 12 (i^{1} i)}{(6 + 4 i) (6 - 4 i)}$

Step 2.2.2.1.4.3

Raise

i

$i$ to the power of

1

$1$ .

\frac{12 - 8 i - 18 i + 12 (i^{1} i^{1})}{(6 + 4 i) (6 - 4 i)}

$\frac{12 - 8 i - 18 i + 12 (i^{1} i^{1})}{(6 + 4 i) (6 - 4 i)}$

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{12 - 8 i - 18 i + 12 i^{1 + 1}}{(6 + 4 i) (6 - 4 i)}

$\frac{12 - 8 i - 18 i + 12 i^{1 + 1}}{(6 + 4 i) (6 - 4 i)}$

Step 2.2.2.1.4.5

Add

1

$1$ and

1

$1$ .

\frac{12 - 8 i - 18 i + 12 i^{2}}{(6 + 4 i) (6 - 4 i)}

$\frac{12 - 8 i - 18 i + 12 i^{2}}{(6 + 4 i) (6 - 4 i)}$

\frac{12 - 8 i - 18 i + 12 i^{2}}{(6 + 4 i) (6 - 4 i)}

$\frac{12 - 8 i - 18 i + 12 i^{2}}{(6 + 4 i) (6 - 4 i)}$

Step 2.2.2.1.5

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{12 - 8 i - 18 i + 12 \cdot - 1}{(6 + 4 i) (6 - 4 i)}

$\frac{12 - 8 i - 18 i + 12 \cdot - 1}{(6 + 4 i) (6 - 4 i)}$

Step 2.2.2.1.6

Multiply

12

$12$ by

- 1

$- 1$ .

\frac{12 - 8 i - 18 i - 12}{(6 + 4 i) (6 - 4 i)}

$\frac{12 - 8 i - 18 i - 12}{(6 + 4 i) (6 - 4 i)}$

\frac{12 - 8 i - 18 i - 12}{(6 + 4 i) (6 - 4 i)}

$\frac{12 - 8 i - 18 i - 12}{(6 + 4 i) (6 - 4 i)}$

Step 2.2.2.2

Subtract

12

$12$ from

12

$12$ .

\frac{0 - 8 i - 18 i}{(6 + 4 i) (6 - 4 i)}

$\frac{0 - 8 i - 18 i}{(6 + 4 i) (6 - 4 i)}$

Step 2.2.2.3

Subtract

8 i

$8 i$ from

0

$0$ .

\frac{- 8 i - 18 i}{(6 + 4 i) (6 - 4 i)}

$\frac{- 8 i - 18 i}{(6 + 4 i) (6 - 4 i)}$

Step 2.2.2.4

Subtract

18 i

$18 i$ from

- 8 i

$- 8 i$ .

\frac{- 26 i}{(6 + 4 i) (6 - 4 i)}

$\frac{- 26 i}{(6 + 4 i) (6 - 4 i)}$

\frac{- 26 i}{(6 + 4 i) (6 - 4 i)}

$\frac{- 26 i}{(6 + 4 i) (6 - 4 i)}$

\frac{- 26 i}{(6 + 4 i) (6 - 4 i)}

$\frac{- 26 i}{(6 + 4 i) (6 - 4 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

(6 + 4 i) (6 - 4 i)

$(6 + 4 i) (6 - 4 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

\frac{- 26 i}{6 (6 - 4 i) + 4 i (6 - 4 i)}

$\frac{- 26 i}{6 (6 - 4 i) + 4 i (6 - 4 i)}$

Step 2.3.1.2

Apply the distributive property.

\frac{- 26 i}{6 \cdot 6 + 6 (- 4 i) + 4 i (6 - 4 i)}

$\frac{- 26 i}{6 \cdot 6 + 6 (- 4 i) + 4 i (6 - 4 i)}$

Step 2.3.1.3

Apply the distributive property.

\frac{- 26 i}{6 \cdot 6 + 6 (- 4 i) + 4 i \cdot 6 + 4 i (- 4 i)}

$\frac{- 26 i}{6 \cdot 6 + 6 (- 4 i) + 4 i \cdot 6 + 4 i (- 4 i)}$

\frac{- 26 i}{6 \cdot 6 + 6 (- 4 i) + 4 i \cdot 6 + 4 i (- 4 i)}

$\frac{- 26 i}{6 \cdot 6 + 6 (- 4 i) + 4 i \cdot 6 + 4 i (- 4 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

6

$6$ by

6

$6$ .

\frac{- 26 i}{36 + 6 (- 4 i) + 4 i \cdot 6 + 4 i (- 4 i)}

$\frac{- 26 i}{36 + 6 (- 4 i) + 4 i \cdot 6 + 4 i (- 4 i)}$

Step 2.3.2.2

Multiply

- 4

$- 4$ by

6

$6$ .

\frac{- 26 i}{36 - 24 i + 4 i \cdot 6 + 4 i (- 4 i)}

$\frac{- 26 i}{36 - 24 i + 4 i \cdot 6 + 4 i (- 4 i)}$

Step 2.3.2.3

Multiply

6

$6$ by

4

$4$ .

\frac{- 26 i}{36 - 24 i + 24 i + 4 i (- 4 i)}

$\frac{- 26 i}{36 - 24 i + 24 i + 4 i (- 4 i)}$

Step 2.3.2.4

Multiply

- 4

$- 4$ by

4

$4$ .

\frac{- 26 i}{36 - 24 i + 24 i - 16 i i}

$\frac{- 26 i}{36 - 24 i + 24 i - 16 i i}$

Step 2.3.2.5

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 26 i}{36 - 24 i + 24 i - 16 (i^{1} i)}

$\frac{- 26 i}{36 - 24 i + 24 i - 16 (i^{1} i)}$

Step 2.3.2.6

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 26 i}{36 - 24 i + 24 i - 16 (i^{1} i^{1})}

$\frac{- 26 i}{36 - 24 i + 24 i - 16 (i^{1} i^{1})}$

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{- 26 i}{36 - 24 i + 24 i - 16 i^{1 + 1}}

$\frac{- 26 i}{36 - 24 i + 24 i - 16 i^{1 + 1}}$

Step 2.3.2.8

Add

1

$1$ and

1

$1$ .

\frac{- 26 i}{36 - 24 i + 24 i - 16 i^{2}}

$\frac{- 26 i}{36 - 24 i + 24 i - 16 i^{2}}$

Step 2.3.2.9

Add

- 24 i

$- 24 i$ and

24 i

$24 i$ .

\frac{- 26 i}{36 + 0 - 16 i^{2}}

$\frac{- 26 i}{36 + 0 - 16 i^{2}}$

Step 2.3.2.10

Add

36

$36$ and

0

$0$ .

\frac{- 26 i}{36 - 16 i^{2}}

$\frac{- 26 i}{36 - 16 i^{2}}$

\frac{- 26 i}{36 - 16 i^{2}}

$\frac{- 26 i}{36 - 16 i^{2}}$

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{- 26 i}{36 - 16 \cdot - 1}

$\frac{- 26 i}{36 - 16 \cdot - 1}$

Step 2.3.3.2

Multiply

- 16

$- 16$ by

- 1

$- 1$ .

\frac{- 26 i}{36 + 16}

$\frac{- 26 i}{36 + 16}$

\frac{- 26 i}{36 + 16}

$\frac{- 26 i}{36 + 16}$

Step 2.3.4

Add

36

$36$ and

16

$16$ .

\frac{- 26 i}{52}

$\frac{- 26 i}{52}$

\frac{- 26 i}{52}

$\frac{- 26 i}{52}$

\frac{- 26 i}{52}

$\frac{- 26 i}{52}$

Step 3

Cancel the common factor of

- 26

$- 26$ and

52

$52$ .

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Step 3.1

Factor

26

$26$ out of

- 26 i

$- 26 i$ .

\frac{26 (- i)}{52}

$\frac{26 (- i)}{52}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor

26

$26$ out of

52

$52$ .

\frac{26 (- i)}{26 (2)}

$\frac{26 (- i)}{26 (2)}$

Step 3.2.2

Cancel the common factor.

$\frac{26 (- i)}{26 \cdot 2}$

Step 3.2.3

Rewrite the expression.

$\frac{- i}{2}$

Step 4

Move the negative in front of the fraction.

$- \frac{i}{2}$