Basic Math Examples

\frac{8 + 6 i}{- 3 + i}

$\frac{8 + 6 i}{- 3 + i}$

Step 1

Multiply the numerator and denominator of

\frac{8 + 6 i}{- 3 + 1 i}

$\frac{8 + 6 i}{- 3 + 1 i}$ by the conjugate of

- 3 + 1 i

$- 3 + 1 i$ to make the denominator real.

\frac{8 + 6 i}{- 3 + 1 i} \cdot \frac{- 3 - i}{- 3 - i}

$\frac{8 + 6 i}{- 3 + 1 i} \cdot \frac{- 3 - i}{- 3 - i}$

Step 2

Multiply.

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Step 2.1

Combine.

\frac{(8 + 6 i) (- 3 - i)}{(- 3 + 1 i) (- 3 - i)}

$\frac{(8 + 6 i) (- 3 - i)}{(- 3 + 1 i) (- 3 - i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(8 + 6 i) (- 3 - i)

$(8 + 6 i) (- 3 - i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

\frac{8 (- 3 - i) + 6 i (- 3 - i)}{(- 3 + 1 i) (- 3 - i)}

$\frac{8 (- 3 - i) + 6 i (- 3 - i)}{(- 3 + 1 i) (- 3 - i)}$

Step 2.2.1.2

Apply the distributive property.

\frac{8 \cdot - 3 + 8 (- i) + 6 i (- 3 - i)}{(- 3 + 1 i) (- 3 - i)}

$\frac{8 \cdot - 3 + 8 (- i) + 6 i (- 3 - i)}{(- 3 + 1 i) (- 3 - i)}$

Step 2.2.1.3

Apply the distributive property.

\frac{8 \cdot - 3 + 8 (- i) + 6 i \cdot - 3 + 6 i (- i)}{(- 3 + 1 i) (- 3 - i)}

$\frac{8 \cdot - 3 + 8 (- i) + 6 i \cdot - 3 + 6 i (- i)}{(- 3 + 1 i) (- 3 - i)}$

\frac{8 \cdot - 3 + 8 (- i) + 6 i \cdot - 3 + 6 i (- i)}{(- 3 + 1 i) (- 3 - i)}

$\frac{8 \cdot - 3 + 8 (- i) + 6 i \cdot - 3 + 6 i (- i)}{(- 3 + 1 i) (- 3 - i)}$

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

8

$8$ by

- 3

$- 3$ .

\frac{- 24 + 8 (- i) + 6 i \cdot - 3 + 6 i (- i)}{(- 3 + 1 i) (- 3 - i)}

$\frac{- 24 + 8 (- i) + 6 i \cdot - 3 + 6 i (- i)}{(- 3 + 1 i) (- 3 - i)}$

Step 2.2.2.1.2

Multiply

- 1

$- 1$ by

8

$8$ .

\frac{- 24 - 8 i + 6 i \cdot - 3 + 6 i (- i)}{(- 3 + 1 i) (- 3 - i)}

$\frac{- 24 - 8 i + 6 i \cdot - 3 + 6 i (- i)}{(- 3 + 1 i) (- 3 - i)}$

Step 2.2.2.1.3

Multiply

- 3

$- 3$ by

6

$6$ .

\frac{- 24 - 8 i - 18 i + 6 i (- i)}{(- 3 + 1 i) (- 3 - i)}

$\frac{- 24 - 8 i - 18 i + 6 i (- i)}{(- 3 + 1 i) (- 3 - i)}$

Step 2.2.2.1.4

Multiply

6 i (- i)

$6 i (- i)$ .

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Step 2.2.2.1.4.1

Multiply

- 1

$- 1$ by

6

$6$ .

\frac{- 24 - 8 i - 18 i - 6 i i}{(- 3 + 1 i) (- 3 - i)}

$\frac{- 24 - 8 i - 18 i - 6 i i}{(- 3 + 1 i) (- 3 - i)}$

Step 2.2.2.1.4.2

Raise

i

$i$ to the power of

$1$ .

$\frac{- 24 - 8 i - 18 i - 6 (i^{1} i)}{(- 3 + 1 i) (- 3 - i)}$

Step 2.2.2.1.4.3

Raise

$i$ to the power of

$1$ .

$\frac{- 24 - 8 i - 18 i - 6 (i^{1} i^{1})}{(- 3 + 1 i) (- 3 - i)}$

Step 2.2.2.1.4.4

Use the power rule

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

$\frac{- 24 - 8 i - 18 i - 6 i^{1 + 1}}{(- 3 + 1 i) (- 3 - i)}$

Step 2.2.2.1.4.5

Add

$1$ and

$1$ .

$\frac{- 24 - 8 i - 18 i - 6 i^{2}}{(- 3 + 1 i) (- 3 - i)}$

Step 2.2.2.1.5

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{- 24 - 8 i - 18 i - 6 \cdot - 1}{(- 3 + 1 i) (- 3 - i)}$

Step 2.2.2.1.6

Multiply

$- 6$ by

$- 1$ .

$\frac{- 24 - 8 i - 18 i + 6}{(- 3 + 1 i) (- 3 - i)}$

Step 2.2.2.2

Add

$- 24$ and

$6$ .

$\frac{- 18 - 8 i - 18 i}{(- 3 + 1 i) (- 3 - i)}$

Step 2.2.2.3

Subtract

$18 i$ from

$- 8 i$ .

$\frac{- 18 - 26 i}{(- 3 + 1 i) (- 3 - i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

$(- 3 + 1 i) (- 3 - i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 18 - 26 i}{- 3 (- 3 - i) + 1 i (- 3 - i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 18 - 26 i}{- 3 \cdot - 3 - 3 (- i) + 1 i (- 3 - i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 18 - 26 i}{- 3 \cdot - 3 - 3 (- i) + 1 i \cdot - 3 + 1 i (- i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

$- 3$ by

$- 3$ .

$\frac{- 18 - 26 i}{9 - 3 (- i) + 1 i \cdot - 3 + 1 i (- i)}$

Step 2.3.2.2

Multiply

$- 1$ by

$- 3$ .

$\frac{- 18 - 26 i}{9 + 3 i + 1 i \cdot - 3 + 1 i (- i)}$

Step 2.3.2.3

Multiply

$- 3$ by

$1$ .

$\frac{- 18 - 26 i}{9 + 3 i - 3 i + 1 i (- i)}$

Step 2.3.2.4

Multiply

$- 1$ by

$1$ .

$\frac{- 18 - 26 i}{9 + 3 i - 3 i - i i}$

Step 2.3.2.5

Raise

$i$ to the power of

$1$ .

$\frac{- 18 - 26 i}{9 + 3 i - 3 i - (i^{1} i)}$

Step 2.3.2.6

Raise

$i$ to the power of

$1$ .

$\frac{- 18 - 26 i}{9 + 3 i - 3 i - (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

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

$\frac{- 18 - 26 i}{9 + 3 i - 3 i - i^{1 + 1}}$

Step 2.3.2.8

Add

$1$ and

$1$ .

$\frac{- 18 - 26 i}{9 + 3 i - 3 i - i^{2}}$

Step 2.3.2.9

Subtract

$3 i$ from

$3 i$ .

$\frac{- 18 - 26 i}{9 + 0 - i^{2}}$

Step 2.3.2.10

Add

$9$ and

$0$ .

$\frac{- 18 - 26 i}{9 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{- 18 - 26 i}{9 - - 1}$

Step 2.3.3.2

Multiply

$- 1$ by

$- 1$ .

$\frac{- 18 - 26 i}{9 + 1}$

Step 2.3.4

Add

$9$ and

$1$ .

$\frac{- 18 - 26 i}{10}$

Step 3

Cancel the common factor of

$- 18 - 26 i$ and

$10$ .

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Step 3.1

Factor

$2$ out of

$- 18$ .

$\frac{2 (- 9) - 26 i}{10}$

Step 3.2

Factor

$2$ out of

$- 26 i$ .

$\frac{2 (- 9) + 2 (- 13 i)}{10}$

Step 3.3

Factor

$2$ out of

$2 (- 9) + 2 (- 13 i)$ .

$\frac{2 (- 9 - 13 i)}{10}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor

$2$ out of

$10$ .

$\frac{2 (- 9 - 13 i)}{2 \cdot 5}$

Step 3.4.2

Cancel the common factor.

$\frac{2 (- 9 - 13 i)}{2 \cdot 5}$

Step 3.4.3

Rewrite the expression.

$\frac{- 9 - 13 i}{5}$

Step 4

Split the fraction

$\frac{- 9 - 13 i}{5}$ into two fractions.

$\frac{- 9}{5} + \frac{- 13 i}{5}$

Step 5

Simplify each term.

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Step 5.1

Move the negative in front of the fraction.

$- \frac{9}{5} + \frac{- 13 i}{5}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{9}{5} - \frac{13 i}{5}$