Basic Math Examples

\frac{- 3 - 3 i}{4 - 5 i}

$\frac{- 3 - 3 i}{4 - 5 i}$

Step 1

Multiply the numerator and denominator of

\frac{- 3 - 3 i}{4 - 5 i}

$\frac{- 3 - 3 i}{4 - 5 i}$ by the conjugate of

4 - 5 i

$4 - 5 i$ to make the denominator real.

\frac{- 3 - 3 i}{4 - 5 i} \cdot \frac{4 + 5 i}{4 + 5 i}

$\frac{- 3 - 3 i}{4 - 5 i} \cdot \frac{4 + 5 i}{4 + 5 i}$

Step 2

Multiply.

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Step 2.1

Combine.

\frac{(- 3 - 3 i) (4 + 5 i)}{(4 - 5 i) (4 + 5 i)}

$\frac{(- 3 - 3 i) (4 + 5 i)}{(4 - 5 i) (4 + 5 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(- 3 - 3 i) (4 + 5 i)

$(- 3 - 3 i) (4 + 5 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

\frac{- 3 (4 + 5 i) - 3 i (4 + 5 i)}{(4 - 5 i) (4 + 5 i)}

$\frac{- 3 (4 + 5 i) - 3 i (4 + 5 i)}{(4 - 5 i) (4 + 5 i)}$

Step 2.2.1.2

Apply the distributive property.

\frac{- 3 \cdot 4 - 3 (5 i) - 3 i (4 + 5 i)}{(4 - 5 i) (4 + 5 i)}

$\frac{- 3 \cdot 4 - 3 (5 i) - 3 i (4 + 5 i)}{(4 - 5 i) (4 + 5 i)}$

Step 2.2.1.3

Apply the distributive property.

\frac{- 3 \cdot 4 - 3 (5 i) - 3 i \cdot 4 - 3 i (5 i)}{(4 - 5 i) (4 + 5 i)}

$\frac{- 3 \cdot 4 - 3 (5 i) - 3 i \cdot 4 - 3 i (5 i)}{(4 - 5 i) (4 + 5 i)}$

\frac{- 3 \cdot 4 - 3 (5 i) - 3 i \cdot 4 - 3 i (5 i)}{(4 - 5 i) (4 + 5 i)}

$\frac{- 3 \cdot 4 - 3 (5 i) - 3 i \cdot 4 - 3 i (5 i)}{(4 - 5 i) (4 + 5 i)}$

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

- 3

$- 3$ by

4

$4$ .

\frac{- 12 - 3 (5 i) - 3 i \cdot 4 - 3 i (5 i)}{(4 - 5 i) (4 + 5 i)}

$\frac{- 12 - 3 (5 i) - 3 i \cdot 4 - 3 i (5 i)}{(4 - 5 i) (4 + 5 i)}$

Step 2.2.2.1.2

Multiply

5

$5$ by

- 3

$- 3$ .

\frac{- 12 - 15 i - 3 i \cdot 4 - 3 i (5 i)}{(4 - 5 i) (4 + 5 i)}

$\frac{- 12 - 15 i - 3 i \cdot 4 - 3 i (5 i)}{(4 - 5 i) (4 + 5 i)}$

Step 2.2.2.1.3

Multiply

4

$4$ by

- 3

$- 3$ .

\frac{- 12 - 15 i - 12 i - 3 i (5 i)}{(4 - 5 i) (4 + 5 i)}

$\frac{- 12 - 15 i - 12 i - 3 i (5 i)}{(4 - 5 i) (4 + 5 i)}$

Step 2.2.2.1.4

Multiply

- 3 i (5 i)

$- 3 i (5 i)$ .

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Step 2.2.2.1.4.1

Multiply

5

$5$ by

- 3

$- 3$ .

\frac{- 12 - 15 i - 12 i - 15 i i}{(4 - 5 i) (4 + 5 i)}

$\frac{- 12 - 15 i - 12 i - 15 i i}{(4 - 5 i) (4 + 5 i)}$

Step 2.2.2.1.4.2

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 12 - 15 i - 12 i - 15 (i^{1} i)}{(4 - 5 i) (4 + 5 i)}

$\frac{- 12 - 15 i - 12 i - 15 (i^{1} i)}{(4 - 5 i) (4 + 5 i)}$

Step 2.2.2.1.4.3

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 12 - 15 i - 12 i - 15 (i^{1} i^{1})}{(4 - 5 i) (4 + 5 i)}

$\frac{- 12 - 15 i - 12 i - 15 (i^{1} i^{1})}{(4 - 5 i) (4 + 5 i)}$

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 - 15 i - 12 i - 15 i^{1 + 1}}{(4 - 5 i) (4 + 5 i)}

$\frac{- 12 - 15 i - 12 i - 15 i^{1 + 1}}{(4 - 5 i) (4 + 5 i)}$

Step 2.2.2.1.4.5

Add

1

$1$ and

1

$1$ .

\frac{- 12 - 15 i - 12 i - 15 i^{2}}{(4 - 5 i) (4 + 5 i)}

$\frac{- 12 - 15 i - 12 i - 15 i^{2}}{(4 - 5 i) (4 + 5 i)}$

\frac{- 12 - 15 i - 12 i - 15 i^{2}}{(4 - 5 i) (4 + 5 i)}

$\frac{- 12 - 15 i - 12 i - 15 i^{2}}{(4 - 5 i) (4 + 5 i)}$

Step 2.2.2.1.5

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{- 12 - 15 i - 12 i - 15 \cdot - 1}{(4 - 5 i) (4 + 5 i)}

$\frac{- 12 - 15 i - 12 i - 15 \cdot - 1}{(4 - 5 i) (4 + 5 i)}$

Step 2.2.2.1.6

Multiply

- 15

$- 15$ by

- 1

$- 1$ .

\frac{- 12 - 15 i - 12 i + 15}{(4 - 5 i) (4 + 5 i)}

$\frac{- 12 - 15 i - 12 i + 15}{(4 - 5 i) (4 + 5 i)}$

\frac{- 12 - 15 i - 12 i + 15}{(4 - 5 i) (4 + 5 i)}

$\frac{- 12 - 15 i - 12 i + 15}{(4 - 5 i) (4 + 5 i)}$

Step 2.2.2.2

Add

- 12

$- 12$ and

15

$15$ .

\frac{3 - 15 i - 12 i}{(4 - 5 i) (4 + 5 i)}

$\frac{3 - 15 i - 12 i}{(4 - 5 i) (4 + 5 i)}$

Step 2.2.2.3

Subtract

12 i

$12 i$ from

- 15 i

$- 15 i$ .

\frac{3 - 27 i}{(4 - 5 i) (4 + 5 i)}

$\frac{3 - 27 i}{(4 - 5 i) (4 + 5 i)}$

\frac{3 - 27 i}{(4 - 5 i) (4 + 5 i)}

$\frac{3 - 27 i}{(4 - 5 i) (4 + 5 i)}$

\frac{3 - 27 i}{(4 - 5 i) (4 + 5 i)}

$\frac{3 - 27 i}{(4 - 5 i) (4 + 5 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

(4 - 5 i) (4 + 5 i)

$(4 - 5 i) (4 + 5 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

\frac{3 - 27 i}{4 (4 + 5 i) - 5 i (4 + 5 i)}

$\frac{3 - 27 i}{4 (4 + 5 i) - 5 i (4 + 5 i)}$

Step 2.3.1.2

Apply the distributive property.

\frac{3 - 27 i}{4 \cdot 4 + 4 (5 i) - 5 i (4 + 5 i)}

$\frac{3 - 27 i}{4 \cdot 4 + 4 (5 i) - 5 i (4 + 5 i)}$

Step 2.3.1.3

Apply the distributive property.

\frac{3 - 27 i}{4 \cdot 4 + 4 (5 i) - 5 i \cdot 4 - 5 i (5 i)}

$\frac{3 - 27 i}{4 \cdot 4 + 4 (5 i) - 5 i \cdot 4 - 5 i (5 i)}$

\frac{3 - 27 i}{4 \cdot 4 + 4 (5 i) - 5 i \cdot 4 - 5 i (5 i)}

$\frac{3 - 27 i}{4 \cdot 4 + 4 (5 i) - 5 i \cdot 4 - 5 i (5 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

4

$4$ by

4

$4$ .

\frac{3 - 27 i}{16 + 4 (5 i) - 5 i \cdot 4 - 5 i (5 i)}

$\frac{3 - 27 i}{16 + 4 (5 i) - 5 i \cdot 4 - 5 i (5 i)}$

Step 2.3.2.2

Multiply

5

$5$ by

4

$4$ .

\frac{3 - 27 i}{16 + 20 i - 5 i \cdot 4 - 5 i (5 i)}

$\frac{3 - 27 i}{16 + 20 i - 5 i \cdot 4 - 5 i (5 i)}$

Step 2.3.2.3

Multiply

4

$4$ by

- 5

$- 5$ .

\frac{3 - 27 i}{16 + 20 i - 20 i - 5 i (5 i)}

$\frac{3 - 27 i}{16 + 20 i - 20 i - 5 i (5 i)}$

Step 2.3.2.4

Multiply

5

$5$ by

- 5

$- 5$ .

\frac{3 - 27 i}{16 + 20 i - 20 i - 25 i i}

$\frac{3 - 27 i}{16 + 20 i - 20 i - 25 i i}$

Step 2.3.2.5

Raise

$i$ to the power of

$1$ .

$\frac{3 - 27 i}{16 + 20 i - 20 i - 25 (i^{1} i)}$

Step 2.3.2.6

Raise

$i$ to the power of

$1$ .

$\frac{3 - 27 i}{16 + 20 i - 20 i - 25 (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

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

$\frac{3 - 27 i}{16 + 20 i - 20 i - 25 i^{1 + 1}}$

Step 2.3.2.8

Add

$1$ and

$1$ .

$\frac{3 - 27 i}{16 + 20 i - 20 i - 25 i^{2}}$

Step 2.3.2.9

Subtract

$20 i$ from

$20 i$ .

$\frac{3 - 27 i}{16 + 0 - 25 i^{2}}$

Step 2.3.2.10

Add

$16$ and

$0$ .

$\frac{3 - 27 i}{16 - 25 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{3 - 27 i}{16 - 25 \cdot - 1}$

Step 2.3.3.2

Multiply

$- 25$ by

$- 1$ .

$\frac{3 - 27 i}{16 + 25}$

Step 2.3.4

Add

$16$ and

$25$ .

$\frac{3 - 27 i}{41}$

Step 3

Split the fraction

$\frac{3 - 27 i}{41}$ into two fractions.

$\frac{3}{41} + \frac{- 27 i}{41}$

Step 4

Move the negative in front of the fraction.

$\frac{3}{41} - \frac{27 i}{41}$