Basic Math Examples

\frac{6 + 3 i}{- 8 - 4 i}

$\frac{6 + 3 i}{- 8 - 4 i}$

Step 1

Multiply the numerator and denominator of

\frac{6 + 3 i}{- 8 - 4 i}

$\frac{6 + 3 i}{- 8 - 4 i}$ by the conjugate of

- 8 - 4 i

$- 8 - 4 i$ to make the denominator real.

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

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

Step 2

Multiply.

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Step 2.1

Combine.

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

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(6 + 3 i) (- 8 + 4 i)

$(6 + 3 i) (- 8 + 4 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

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

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

Step 2.2.1.2

Apply the distributive property.

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

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

Step 2.2.1.3

Apply the distributive property.

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

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

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

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

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

6

$6$ by

- 8

$- 8$ .

\frac{- 48 + 6 (4 i) + 3 i \cdot - 8 + 3 i (4 i)}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 48 + 6 (4 i) + 3 i \cdot - 8 + 3 i (4 i)}{(- 8 - 4 i) (- 8 + 4 i)}$

Step 2.2.2.1.2

Multiply

4

$4$ by

6

$6$ .

\frac{- 48 + 24 i + 3 i \cdot - 8 + 3 i (4 i)}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 48 + 24 i + 3 i \cdot - 8 + 3 i (4 i)}{(- 8 - 4 i) (- 8 + 4 i)}$

Step 2.2.2.1.3

Multiply

- 8

$- 8$ by

3

$3$ .

\frac{- 48 + 24 i - 24 i + 3 i (4 i)}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 48 + 24 i - 24 i + 3 i (4 i)}{(- 8 - 4 i) (- 8 + 4 i)}$

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{- 48 + 24 i - 24 i + 12 i i}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 48 + 24 i - 24 i + 12 i i}{(- 8 - 4 i) (- 8 + 4 i)}$

Step 2.2.2.1.4.2

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 48 + 24 i - 24 i + 12 (i^{1} i)}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 48 + 24 i - 24 i + 12 (i^{1} i)}{(- 8 - 4 i) (- 8 + 4 i)}$

Step 2.2.2.1.4.3

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 48 + 24 i - 24 i + 12 (i^{1} i^{1})}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 48 + 24 i - 24 i + 12 (i^{1} i^{1})}{(- 8 - 4 i) (- 8 + 4 i)}$

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{- 48 + 24 i - 24 i + 12 i^{1 + 1}}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 48 + 24 i - 24 i + 12 i^{1 + 1}}{(- 8 - 4 i) (- 8 + 4 i)}$

Step 2.2.2.1.4.5

Add

1

$1$ and

1

$1$ .

\frac{- 48 + 24 i - 24 i + 12 i^{2}}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 48 + 24 i - 24 i + 12 i^{2}}{(- 8 - 4 i) (- 8 + 4 i)}$

\frac{- 48 + 24 i - 24 i + 12 i^{2}}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 48 + 24 i - 24 i + 12 i^{2}}{(- 8 - 4 i) (- 8 + 4 i)}$

Step 2.2.2.1.5

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{- 48 + 24 i - 24 i + 12 \cdot - 1}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 48 + 24 i - 24 i + 12 \cdot - 1}{(- 8 - 4 i) (- 8 + 4 i)}$

Step 2.2.2.1.6

Multiply

12

$12$ by

- 1

$- 1$ .

\frac{- 48 + 24 i - 24 i - 12}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 48 + 24 i - 24 i - 12}{(- 8 - 4 i) (- 8 + 4 i)}$

\frac{- 48 + 24 i - 24 i - 12}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 48 + 24 i - 24 i - 12}{(- 8 - 4 i) (- 8 + 4 i)}$

Step 2.2.2.2

Subtract

12

$12$ from

- 48

$- 48$ .

\frac{- 60 + 24 i - 24 i}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 60 + 24 i - 24 i}{(- 8 - 4 i) (- 8 + 4 i)}$

Step 2.2.2.3

Subtract

24 i

$24 i$ from

24 i

$24 i$ .

\frac{- 60 + 0}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 60 + 0}{(- 8 - 4 i) (- 8 + 4 i)}$

Step 2.2.2.4

Add

- 60

$- 60$ and

0

$0$ .

\frac{- 60}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 60}{(- 8 - 4 i) (- 8 + 4 i)}$

\frac{- 60}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 60}{(- 8 - 4 i) (- 8 + 4 i)}$

\frac{- 60}{(- 8 - 4 i) (- 8 + 4 i)}

$\frac{- 60}{(- 8 - 4 i) (- 8 + 4 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

(- 8 - 4 i) (- 8 + 4 i)

$(- 8 - 4 i) (- 8 + 4 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

\frac{- 60}{- 8 (- 8 + 4 i) - 4 i (- 8 + 4 i)}

$\frac{- 60}{- 8 (- 8 + 4 i) - 4 i (- 8 + 4 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 60}{- 8 \cdot - 8 - 8 (4 i) - 4 i (- 8 + 4 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 60}{- 8 \cdot - 8 - 8 (4 i) - 4 i \cdot - 8 - 4 i (4 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

$- 8$ by

$- 8$ .

$\frac{- 60}{64 - 8 (4 i) - 4 i \cdot - 8 - 4 i (4 i)}$

Step 2.3.2.2

Multiply

$4$ by

$- 8$ .

$\frac{- 60}{64 - 32 i - 4 i \cdot - 8 - 4 i (4 i)}$

Step 2.3.2.3

Multiply

$- 8$ by

$- 4$ .

$\frac{- 60}{64 - 32 i + 32 i - 4 i (4 i)}$

Step 2.3.2.4

Multiply

$4$ by

$- 4$ .

$\frac{- 60}{64 - 32 i + 32 i - 16 i i}$

Step 2.3.2.5

Raise

$i$ to the power of

$1$ .

$\frac{- 60}{64 - 32 i + 32 i - 16 (i^{1} i)}$

Step 2.3.2.6

Raise

$i$ to the power of

$1$ .

$\frac{- 60}{64 - 32 i + 32 i - 16 (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

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

$\frac{- 60}{64 - 32 i + 32 i - 16 i^{1 + 1}}$

Step 2.3.2.8

Add

$1$ and

$1$ .

$\frac{- 60}{64 - 32 i + 32 i - 16 i^{2}}$

Step 2.3.2.9

Add

$- 32 i$ and

$32 i$ .

$\frac{- 60}{64 + 0 - 16 i^{2}}$

Step 2.3.2.10

Add

$64$ and

$0$ .

$\frac{- 60}{64 - 16 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{- 60}{64 - 16 \cdot - 1}$

Step 2.3.3.2

Multiply

$- 16$ by

$- 1$ .

$\frac{- 60}{64 + 16}$

Step 2.3.4

Add

$64$ and

$16$ .

$\frac{- 60}{80}$

Step 3

Cancel the common factor of

$- 60$ and

$80$ .

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Step 3.1

Factor

$20$ out of

$- 60$ .

$\frac{20 (- 3)}{80}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor

$20$ out of

$80$ .

$\frac{20 \cdot - 3}{20 \cdot 4}$

Step 3.2.2

Cancel the common factor.

$\frac{20 \cdot - 3}{20 \cdot 4}$

Step 3.2.3

Rewrite the expression.

$\frac{- 3}{4}$

Step 4

Move the negative in front of the fraction.

$- \frac{3}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{4}$

Decimal Form:

$- 0.75$