Basic Math Examples

\frac{6 - 5 i}{6 + 3 i}

$\frac{6 - 5 i}{6 + 3 i}$

Step 1

Multiply the numerator and denominator of

\frac{6 - 5 i}{6 + 3 i}

$\frac{6 - 5 i}{6 + 3 i}$ by the conjugate of

6 + 3 i

$6 + 3 i$ to make the denominator real.

\frac{6 - 5 i}{6 + 3 i} \cdot \frac{6 - 3 i}{6 - 3 i}

$\frac{6 - 5 i}{6 + 3 i} \cdot \frac{6 - 3 i}{6 - 3 i}$

Step 2

Multiply.

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Step 2.1

Combine.

\frac{(6 - 5 i) (6 - 3 i)}{(6 + 3 i) (6 - 3 i)}

$\frac{(6 - 5 i) (6 - 3 i)}{(6 + 3 i) (6 - 3 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(6 - 5 i) (6 - 3 i)

$(6 - 5 i) (6 - 3 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

\frac{6 (6 - 3 i) - 5 i (6 - 3 i)}{(6 + 3 i) (6 - 3 i)}

$\frac{6 (6 - 3 i) - 5 i (6 - 3 i)}{(6 + 3 i) (6 - 3 i)}$

Step 2.2.1.2

Apply the distributive property.

\frac{6 \cdot 6 + 6 (- 3 i) - 5 i (6 - 3 i)}{(6 + 3 i) (6 - 3 i)}

$\frac{6 \cdot 6 + 6 (- 3 i) - 5 i (6 - 3 i)}{(6 + 3 i) (6 - 3 i)}$

Step 2.2.1.3

Apply the distributive property.

\frac{6 \cdot 6 + 6 (- 3 i) - 5 i \cdot 6 - 5 i (- 3 i)}{(6 + 3 i) (6 - 3 i)}

$\frac{6 \cdot 6 + 6 (- 3 i) - 5 i \cdot 6 - 5 i (- 3 i)}{(6 + 3 i) (6 - 3 i)}$

\frac{6 \cdot 6 + 6 (- 3 i) - 5 i \cdot 6 - 5 i (- 3 i)}{(6 + 3 i) (6 - 3 i)}

$\frac{6 \cdot 6 + 6 (- 3 i) - 5 i \cdot 6 - 5 i (- 3 i)}{(6 + 3 i) (6 - 3 i)}$

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

6

$6$ by

6

$6$ .

\frac{36 + 6 (- 3 i) - 5 i \cdot 6 - 5 i (- 3 i)}{(6 + 3 i) (6 - 3 i)}

$\frac{36 + 6 (- 3 i) - 5 i \cdot 6 - 5 i (- 3 i)}{(6 + 3 i) (6 - 3 i)}$

Step 2.2.2.1.2

Multiply

- 3

$- 3$ by

6

$6$ .

\frac{36 - 18 i - 5 i \cdot 6 - 5 i (- 3 i)}{(6 + 3 i) (6 - 3 i)}

$\frac{36 - 18 i - 5 i \cdot 6 - 5 i (- 3 i)}{(6 + 3 i) (6 - 3 i)}$

Step 2.2.2.1.3

Multiply

6

$6$ by

- 5

$- 5$ .

\frac{36 - 18 i - 30 i - 5 i (- 3 i)}{(6 + 3 i) (6 - 3 i)}

$\frac{36 - 18 i - 30 i - 5 i (- 3 i)}{(6 + 3 i) (6 - 3 i)}$

Step 2.2.2.1.4

Multiply

- 5 i (- 3 i)

$- 5 i (- 3 i)$ .

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Step 2.2.2.1.4.1

Multiply

- 3

$- 3$ by

- 5

$- 5$ .

\frac{36 - 18 i - 30 i + 15 i i}{(6 + 3 i) (6 - 3 i)}

$\frac{36 - 18 i - 30 i + 15 i i}{(6 + 3 i) (6 - 3 i)}$

Step 2.2.2.1.4.2

Raise

i

$i$ to the power of

1

$1$ .

\frac{36 - 18 i - 30 i + 15 (i^{1} i)}{(6 + 3 i) (6 - 3 i)}

$\frac{36 - 18 i - 30 i + 15 (i^{1} i)}{(6 + 3 i) (6 - 3 i)}$

Step 2.2.2.1.4.3

Raise

i

$i$ to the power of

1

$1$ .

\frac{36 - 18 i - 30 i + 15 (i^{1} i^{1})}{(6 + 3 i) (6 - 3 i)}

$\frac{36 - 18 i - 30 i + 15 (i^{1} i^{1})}{(6 + 3 i) (6 - 3 i)}$

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{36 - 18 i - 30 i + 15 i^{1 + 1}}{(6 + 3 i) (6 - 3 i)}

$\frac{36 - 18 i - 30 i + 15 i^{1 + 1}}{(6 + 3 i) (6 - 3 i)}$

Step 2.2.2.1.4.5

Add

$1$ and

$1$ .

$\frac{36 - 18 i - 30 i + 15 i^{2}}{(6 + 3 i) (6 - 3 i)}$

Step 2.2.2.1.5

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{36 - 18 i - 30 i + 15 \cdot - 1}{(6 + 3 i) (6 - 3 i)}$

Step 2.2.2.1.6

Multiply

$15$ by

$- 1$ .

$\frac{36 - 18 i - 30 i - 15}{(6 + 3 i) (6 - 3 i)}$

Step 2.2.2.2

Subtract

$15$ from

$36$ .

$\frac{21 - 18 i - 30 i}{(6 + 3 i) (6 - 3 i)}$

Step 2.2.2.3

Subtract

$30 i$ from

$- 18 i$ .

$\frac{21 - 48 i}{(6 + 3 i) (6 - 3 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{21 - 48 i}{6 (6 - 3 i) + 3 i (6 - 3 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{21 - 48 i}{6 \cdot 6 + 6 (- 3 i) + 3 i (6 - 3 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{21 - 48 i}{6 \cdot 6 + 6 (- 3 i) + 3 i \cdot 6 + 3 i (- 3 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

$6$ by

$6$ .

$\frac{21 - 48 i}{36 + 6 (- 3 i) + 3 i \cdot 6 + 3 i (- 3 i)}$

Step 2.3.2.2

Multiply

$- 3$ by

$6$ .

$\frac{21 - 48 i}{36 - 18 i + 3 i \cdot 6 + 3 i (- 3 i)}$

Step 2.3.2.3

Multiply

$6$ by

$3$ .

$\frac{21 - 48 i}{36 - 18 i + 18 i + 3 i (- 3 i)}$

Step 2.3.2.4

Multiply

$- 3$ by

$3$ .

$\frac{21 - 48 i}{36 - 18 i + 18 i - 9 i i}$

Step 2.3.2.5

Raise

$i$ to the power of

$1$ .

$\frac{21 - 48 i}{36 - 18 i + 18 i - 9 (i^{1} i)}$

Step 2.3.2.6

Raise

$i$ to the power of

$1$ .

$\frac{21 - 48 i}{36 - 18 i + 18 i - 9 (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

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

$\frac{21 - 48 i}{36 - 18 i + 18 i - 9 i^{1 + 1}}$

Step 2.3.2.8

Add

$1$ and

$1$ .

$\frac{21 - 48 i}{36 - 18 i + 18 i - 9 i^{2}}$

Step 2.3.2.9

Add

$- 18 i$ and

$18 i$ .

$\frac{21 - 48 i}{36 + 0 - 9 i^{2}}$

Step 2.3.2.10

Add

$36$ and

$0$ .

$\frac{21 - 48 i}{36 - 9 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{21 - 48 i}{36 - 9 \cdot - 1}$

Step 2.3.3.2

Multiply

$- 9$ by

$- 1$ .

$\frac{21 - 48 i}{36 + 9}$

Step 2.3.4

Add

$36$ and

$9$ .

$\frac{21 - 48 i}{45}$

Step 3

Cancel the common factor of

$21 - 48 i$ and

$45$ .

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Step 3.1

Factor

$3$ out of

$21$ .

$\frac{3 (7) - 48 i}{45}$

Step 3.2

Factor

$3$ out of

$- 48 i$ .

$\frac{3 (7) + 3 (- 16 i)}{45}$

Step 3.3

Factor

$3$ out of

$3 (7) + 3 (- 16 i)$ .

$\frac{3 (7 - 16 i)}{45}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor

$3$ out of

$45$ .

$\frac{3 (7 - 16 i)}{3 \cdot 15}$

Step 3.4.2

Cancel the common factor.

$\frac{3 (7 - 16 i)}{3 \cdot 15}$

Step 3.4.3

Rewrite the expression.

$\frac{7 - 16 i}{15}$

Step 4

Split the fraction

$\frac{7 - 16 i}{15}$ into two fractions.

$\frac{7}{15} + \frac{- 16 i}{15}$

Step 5

Move the negative in front of the fraction.

$\frac{7}{15} - \frac{16 i}{15}$