Basic Math Examples

\frac{- 2 + 2 i}{3 - 8 i}

$\frac{- 2 + 2 i}{3 - 8 i}$

Step 1

Multiply the numerator and denominator of

\frac{- 2 + 2 i}{3 - 8 i}

$\frac{- 2 + 2 i}{3 - 8 i}$ by the conjugate of

3 - 8 i

$3 - 8 i$ to make the denominator real.

\frac{- 2 + 2 i}{3 - 8 i} \cdot \frac{3 + 8 i}{3 + 8 i}

$\frac{- 2 + 2 i}{3 - 8 i} \cdot \frac{3 + 8 i}{3 + 8 i}$

Step 2

Multiply.

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Step 2.1

Combine.

\frac{(- 2 + 2 i) (3 + 8 i)}{(3 - 8 i) (3 + 8 i)}

$\frac{(- 2 + 2 i) (3 + 8 i)}{(3 - 8 i) (3 + 8 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(- 2 + 2 i) (3 + 8 i)

$(- 2 + 2 i) (3 + 8 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

\frac{- 2 (3 + 8 i) + 2 i (3 + 8 i)}{(3 - 8 i) (3 + 8 i)}

$\frac{- 2 (3 + 8 i) + 2 i (3 + 8 i)}{(3 - 8 i) (3 + 8 i)}$

Step 2.2.1.2

Apply the distributive property.

\frac{- 2 \cdot 3 - 2 (8 i) + 2 i (3 + 8 i)}{(3 - 8 i) (3 + 8 i)}

$\frac{- 2 \cdot 3 - 2 (8 i) + 2 i (3 + 8 i)}{(3 - 8 i) (3 + 8 i)}$

Step 2.2.1.3

Apply the distributive property.

\frac{- 2 \cdot 3 - 2 (8 i) + 2 i \cdot 3 + 2 i (8 i)}{(3 - 8 i) (3 + 8 i)}

$\frac{- 2 \cdot 3 - 2 (8 i) + 2 i \cdot 3 + 2 i (8 i)}{(3 - 8 i) (3 + 8 i)}$

\frac{- 2 \cdot 3 - 2 (8 i) + 2 i \cdot 3 + 2 i (8 i)}{(3 - 8 i) (3 + 8 i)}

$\frac{- 2 \cdot 3 - 2 (8 i) + 2 i \cdot 3 + 2 i (8 i)}{(3 - 8 i) (3 + 8 i)}$

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 (8 i) + 2 i \cdot 3 + 2 i (8 i)}{(3 - 8 i) (3 + 8 i)}

$\frac{- 6 - 2 (8 i) + 2 i \cdot 3 + 2 i (8 i)}{(3 - 8 i) (3 + 8 i)}$

Step 2.2.2.1.2

Multiply

8

$8$ by

- 2

$- 2$ .

\frac{- 6 - 16 i + 2 i \cdot 3 + 2 i (8 i)}{(3 - 8 i) (3 + 8 i)}

$\frac{- 6 - 16 i + 2 i \cdot 3 + 2 i (8 i)}{(3 - 8 i) (3 + 8 i)}$

Step 2.2.2.1.3

Multiply

3

$3$ by

2

$2$ .

\frac{- 6 - 16 i + 6 i + 2 i (8 i)}{(3 - 8 i) (3 + 8 i)}

$\frac{- 6 - 16 i + 6 i + 2 i (8 i)}{(3 - 8 i) (3 + 8 i)}$

Step 2.2.2.1.4

Multiply

2 i (8 i)

$2 i (8 i)$ .

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Step 2.2.2.1.4.1

Multiply

8

$8$ by

2

$2$ .

\frac{- 6 - 16 i + 6 i + 16 i i}{(3 - 8 i) (3 + 8 i)}

$\frac{- 6 - 16 i + 6 i + 16 i i}{(3 - 8 i) (3 + 8 i)}$

Step 2.2.2.1.4.2

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 6 - 16 i + 6 i + 16 (i^{1} i)}{(3 - 8 i) (3 + 8 i)}

$\frac{- 6 - 16 i + 6 i + 16 (i^{1} i)}{(3 - 8 i) (3 + 8 i)}$

Step 2.2.2.1.4.3

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 6 - 16 i + 6 i + 16 (i^{1} i^{1})}{(3 - 8 i) (3 + 8 i)}

$\frac{- 6 - 16 i + 6 i + 16 (i^{1} i^{1})}{(3 - 8 i) (3 + 8 i)}$

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 - 16 i + 6 i + 16 i^{1 + 1}}{(3 - 8 i) (3 + 8 i)}

$\frac{- 6 - 16 i + 6 i + 16 i^{1 + 1}}{(3 - 8 i) (3 + 8 i)}$

Step 2.2.2.1.4.5

Add

1

$1$ and

1

$1$ .

\frac{- 6 - 16 i + 6 i + 16 i^{2}}{(3 - 8 i) (3 + 8 i)}

$\frac{- 6 - 16 i + 6 i + 16 i^{2}}{(3 - 8 i) (3 + 8 i)}$

\frac{- 6 - 16 i + 6 i + 16 i^{2}}{(3 - 8 i) (3 + 8 i)}

$\frac{- 6 - 16 i + 6 i + 16 i^{2}}{(3 - 8 i) (3 + 8 i)}$

Step 2.2.2.1.5

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{- 6 - 16 i + 6 i + 16 \cdot - 1}{(3 - 8 i) (3 + 8 i)}

$\frac{- 6 - 16 i + 6 i + 16 \cdot - 1}{(3 - 8 i) (3 + 8 i)}$

Step 2.2.2.1.6

Multiply

16

$16$ by

- 1

$- 1$ .

$\frac{- 6 - 16 i + 6 i - 16}{(3 - 8 i) (3 + 8 i)}$

Step 2.2.2.2

Subtract

$16$ from

$- 6$ .

$\frac{- 22 - 16 i + 6 i}{(3 - 8 i) (3 + 8 i)}$

Step 2.2.2.3

Add

$- 16 i$ and

$6 i$ .

$\frac{- 22 - 10 i}{(3 - 8 i) (3 + 8 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 22 - 10 i}{3 (3 + 8 i) - 8 i (3 + 8 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 22 - 10 i}{3 \cdot 3 + 3 (8 i) - 8 i (3 + 8 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 22 - 10 i}{3 \cdot 3 + 3 (8 i) - 8 i \cdot 3 - 8 i (8 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

$3$ by

$3$ .

$\frac{- 22 - 10 i}{9 + 3 (8 i) - 8 i \cdot 3 - 8 i (8 i)}$

Step 2.3.2.2

Multiply

$8$ by

$3$ .

$\frac{- 22 - 10 i}{9 + 24 i - 8 i \cdot 3 - 8 i (8 i)}$

Step 2.3.2.3

Multiply

$3$ by

$- 8$ .

$\frac{- 22 - 10 i}{9 + 24 i - 24 i - 8 i (8 i)}$

Step 2.3.2.4

Multiply

$8$ by

$- 8$ .

$\frac{- 22 - 10 i}{9 + 24 i - 24 i - 64 i i}$

Step 2.3.2.5

Raise

$i$ to the power of

$1$ .

$\frac{- 22 - 10 i}{9 + 24 i - 24 i - 64 (i^{1} i)}$

Step 2.3.2.6

Raise

$i$ to the power of

$1$ .

$\frac{- 22 - 10 i}{9 + 24 i - 24 i - 64 (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

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

$\frac{- 22 - 10 i}{9 + 24 i - 24 i - 64 i^{1 + 1}}$

Step 2.3.2.8

Add

$1$ and

$1$ .

$\frac{- 22 - 10 i}{9 + 24 i - 24 i - 64 i^{2}}$

Step 2.3.2.9

Subtract

$24 i$ from

$24 i$ .

$\frac{- 22 - 10 i}{9 + 0 - 64 i^{2}}$

Step 2.3.2.10

Add

$9$ and

$0$ .

$\frac{- 22 - 10 i}{9 - 64 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{- 22 - 10 i}{9 - 64 \cdot - 1}$

Step 2.3.3.2

Multiply

$- 64$ by

$- 1$ .

$\frac{- 22 - 10 i}{9 + 64}$

Step 2.3.4

Add

$9$ and

$64$ .

$\frac{- 22 - 10 i}{73}$

Step 3

Split the fraction

$\frac{- 22 - 10 i}{73}$ into two fractions.

$\frac{- 22}{73} + \frac{- 10 i}{73}$

Step 4

Simplify each term.

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Step 4.1

Move the negative in front of the fraction.

$- \frac{22}{73} + \frac{- 10 i}{73}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{22}{73} - \frac{10 i}{73}$