Algebra Examples

\frac{- 1}{4 - 5 i}

$\frac{- 1}{4 - 5 i}$

Step 1

Multiply the numerator and denominator of

\frac{- 1}{4 - 5 i}

$\frac{- 1}{4 - 5 i}$ by the conjugate of

4 - 5 i

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

\frac{- 1}{4 - 5 i} \cdot \frac{4 + 5 i}{4 + 5 i}

$\frac{- 1}{4 - 5 i} \cdot \frac{4 + 5 i}{4 + 5 i}$

Step 2

Multiply.

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Step 2.1

Combine.

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

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

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

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

Step 2.2.2

Multiply

- 1

$- 1$ by

4

$4$ .

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

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

Step 2.2.3

Multiply

5

$5$ by

- 1

$- 1$ .

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

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

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

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

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{- 4 - 5 i}{4 (4 + 5 i) - 5 i (4 + 5 i)}

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

Step 2.3.1.2

Apply the distributive property.

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

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

Step 2.3.1.3

Apply the distributive property.

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

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

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

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

4

$4$ by

4

$4$ .

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

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

Step 2.3.2.2

Multiply

5

$5$ by

4

$4$ .

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

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

Step 2.3.2.3

Multiply

4

$4$ by

- 5

$- 5$ .

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

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

Step 2.3.2.4

Multiply

5

$5$ by

- 5

$- 5$ .

\frac{- 4 - 5 i}{16 + 20 i - 20 i - 25 i i}

$\frac{- 4 - 5 i}{16 + 20 i - 20 i - 25 i i}$

Step 2.3.2.5

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 4 - 5 i}{16 + 20 i - 20 i - 25 (i^{1} i)}

$\frac{- 4 - 5 i}{16 + 20 i - 20 i - 25 (i^{1} i)}$

Step 2.3.2.6

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 4 - 5 i}{16 + 20 i - 20 i - 25 (i^{1} i^{1})}

$\frac{- 4 - 5 i}{16 + 20 i - 20 i - 25 (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

a^{m} a^{n} = a^{m + n}

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

\frac{- 4 - 5 i}{16 + 20 i - 20 i - 25 i^{1 + 1}}

$\frac{- 4 - 5 i}{16 + 20 i - 20 i - 25 i^{1 + 1}}$

Step 2.3.2.8

Add

1

$1$ and

1

$1$ .

\frac{- 4 - 5 i}{16 + 20 i - 20 i - 25 i^{2}}

$\frac{- 4 - 5 i}{16 + 20 i - 20 i - 25 i^{2}}$

Step 2.3.2.9

Subtract

20 i

$20 i$ from

20 i

$20 i$ .

\frac{- 4 - 5 i}{16 + 0 - 25 i^{2}}

$\frac{- 4 - 5 i}{16 + 0 - 25 i^{2}}$

Step 2.3.2.10

Add

16

$16$ and

0

$0$ .

\frac{- 4 - 5 i}{16 - 25 i^{2}}

$\frac{- 4 - 5 i}{16 - 25 i^{2}}$

\frac{- 4 - 5 i}{16 - 25 i^{2}}

$\frac{- 4 - 5 i}{16 - 25 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{- 4 - 5 i}{16 - 25 \cdot - 1}

$\frac{- 4 - 5 i}{16 - 25 \cdot - 1}$

Step 2.3.3.2

Multiply

- 25

$- 25$ by

- 1

$- 1$ .

\frac{- 4 - 5 i}{16 + 25}

$\frac{- 4 - 5 i}{16 + 25}$

\frac{- 4 - 5 i}{16 + 25}

$\frac{- 4 - 5 i}{16 + 25}$

Step 2.3.4

Add

16

$16$ and

25

$25$ .

\frac{- 4 - 5 i}{41}

$\frac{- 4 - 5 i}{41}$

\frac{- 4 - 5 i}{41}

$\frac{- 4 - 5 i}{41}$

\frac{- 4 - 5 i}{41}

$\frac{- 4 - 5 i}{41}$

Step 3

Split the fraction

\frac{- 4 - 5 i}{41}

$\frac{- 4 - 5 i}{41}$ into two fractions.

\frac{- 4}{41} + \frac{- 5 i}{41}

$\frac{- 4}{41} + \frac{- 5 i}{41}$

Step 4

Simplify each term.

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Step 4.1

Move the negative in front of the fraction.

- \frac{4}{41} + \frac{- 5 i}{41}

$- \frac{4}{41} + \frac{- 5 i}{41}$

Step 4.2

Move the negative in front of the fraction.

- \frac{4}{41} - \frac{5 i}{41}

$- \frac{4}{41} - \frac{5 i}{41}$

- \frac{4}{41} - \frac{5 i}{41}

$- \frac{4}{41} - \frac{5 i}{41}$

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