Algebra Examples

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

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

Step 1

Multiply the numerator and denominator of

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

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

- 1 + 5 i

$- 1 + 5 i$ to make the denominator real.

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

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

Step 2

Multiply.

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Step 2.1

Combine.

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

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(i + 1) (- 1 - 5 i)

$(i + 1) (- 1 - 5 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

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

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

Step 2.2.1.2

Apply the distributive property.

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

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

Step 2.2.1.3

Apply the distributive property.

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

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

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

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

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

Move

- 1

$- 1$ to the left of

i

$i$ .

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

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

Step 2.2.2.1.2

Rewrite

- 1 i

$- 1 i$ as

- i

$- i$ .

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

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

Step 2.2.2.1.3

Multiply

i (- 5 i)

$i (- 5 i)$ .

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Step 2.2.2.1.3.1

Raise

i

$i$ to the power of

1

$1$ .

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

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

Step 2.2.2.1.3.2

Raise

i

$i$ to the power of

1

$1$ .

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

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

Step 2.2.2.1.3.3

Use the power rule

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

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

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

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

Step 2.2.2.1.3.4

Add

1

$1$ and

1

$1$ .

\frac{- i - 5 i^{2} + 1 \cdot - 1 + 1 (- 5 i)}{(- 1 + 5 i) (- 1 - 5 i)}

$\frac{- i - 5 i^{2} + 1 \cdot - 1 + 1 (- 5 i)}{(- 1 + 5 i) (- 1 - 5 i)}$

\frac{- i - 5 i^{2} + 1 \cdot - 1 + 1 (- 5 i)}{(- 1 + 5 i) (- 1 - 5 i)}

$\frac{- i - 5 i^{2} + 1 \cdot - 1 + 1 (- 5 i)}{(- 1 + 5 i) (- 1 - 5 i)}$

Step 2.2.2.1.4

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

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

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

Step 2.2.2.1.5

Multiply

- 5

$- 5$ by

- 1

$- 1$ .

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

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

Step 2.2.2.1.6

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

Step 2.2.2.1.7

Multiply

- 5 i

$- 5 i$ by

1

$1$ .

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

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

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

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

Step 2.2.2.2

Subtract

5 i

$5 i$ from

- i

$- i$ .

\frac{5 - 1 - 6 i}{(- 1 + 5 i) (- 1 - 5 i)}

$\frac{5 - 1 - 6 i}{(- 1 + 5 i) (- 1 - 5 i)}$

Step 2.2.2.3

Subtract

1

$1$ from

5

$5$ .

\frac{4 - 6 i}{(- 1 + 5 i) (- 1 - 5 i)}

$\frac{4 - 6 i}{(- 1 + 5 i) (- 1 - 5 i)}$

\frac{4 - 6 i}{(- 1 + 5 i) (- 1 - 5 i)}

$\frac{4 - 6 i}{(- 1 + 5 i) (- 1 - 5 i)}$

\frac{4 - 6 i}{(- 1 + 5 i) (- 1 - 5 i)}

$\frac{4 - 6 i}{(- 1 + 5 i) (- 1 - 5 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

(- 1 + 5 i) (- 1 - 5 i)

$(- 1 + 5 i) (- 1 - 5 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

\frac{4 - 6 i}{- 1 (- 1 - 5 i) + 5 i (- 1 - 5 i)}

$\frac{4 - 6 i}{- 1 (- 1 - 5 i) + 5 i (- 1 - 5 i)}$

Step 2.3.1.2

Apply the distributive property.

\frac{4 - 6 i}{- 1 \cdot - 1 - 1 (- 5 i) + 5 i (- 1 - 5 i)}

$\frac{4 - 6 i}{- 1 \cdot - 1 - 1 (- 5 i) + 5 i (- 1 - 5 i)}$

Step 2.3.1.3

Apply the distributive property.

\frac{4 - 6 i}{- 1 \cdot - 1 - 1 (- 5 i) + 5 i \cdot - 1 + 5 i (- 5 i)}

$\frac{4 - 6 i}{- 1 \cdot - 1 - 1 (- 5 i) + 5 i \cdot - 1 + 5 i (- 5 i)}$

\frac{4 - 6 i}{- 1 \cdot - 1 - 1 (- 5 i) + 5 i \cdot - 1 + 5 i (- 5 i)}

$\frac{4 - 6 i}{- 1 \cdot - 1 - 1 (- 5 i) + 5 i \cdot - 1 + 5 i (- 5 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{4 - 6 i}{1 - 1 (- 5 i) + 5 i \cdot - 1 + 5 i (- 5 i)}

$\frac{4 - 6 i}{1 - 1 (- 5 i) + 5 i \cdot - 1 + 5 i (- 5 i)}$

Step 2.3.2.2

Multiply

- 5

$- 5$ by

- 1

$- 1$ .

\frac{4 - 6 i}{1 + 5 i + 5 i \cdot - 1 + 5 i (- 5 i)}

$\frac{4 - 6 i}{1 + 5 i + 5 i \cdot - 1 + 5 i (- 5 i)}$

Step 2.3.2.3

Multiply

- 1

$- 1$ by

5

$5$ .

\frac{4 - 6 i}{1 + 5 i - 5 i + 5 i (- 5 i)}

$\frac{4 - 6 i}{1 + 5 i - 5 i + 5 i (- 5 i)}$

Step 2.3.2.4

Multiply

- 5

$- 5$ by

5

$5$ .

\frac{4 - 6 i}{1 + 5 i - 5 i - 25 i i}

$\frac{4 - 6 i}{1 + 5 i - 5 i - 25 i i}$

Step 2.3.2.5

Raise

i

$i$ to the power of

1

$1$ .

\frac{4 - 6 i}{1 + 5 i - 5 i - 25 (i^{1} i)}

$\frac{4 - 6 i}{1 + 5 i - 5 i - 25 (i^{1} i)}$

Step 2.3.2.6

Raise

i

$i$ to the power of

1

$1$ .

\frac{4 - 6 i}{1 + 5 i - 5 i - 25 (i^{1} i^{1})}

$\frac{4 - 6 i}{1 + 5 i - 5 i - 25 (i^{1} i^{1})}$

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 - 6 i}{1 + 5 i - 5 i - 25 i^{1 + 1}}

$\frac{4 - 6 i}{1 + 5 i - 5 i - 25 i^{1 + 1}}$

Step 2.3.2.8

Add

1

$1$ and

1

$1$ .

\frac{4 - 6 i}{1 + 5 i - 5 i - 25 i^{2}}

$\frac{4 - 6 i}{1 + 5 i - 5 i - 25 i^{2}}$

Step 2.3.2.9

Subtract

5 i

$5 i$ from

5 i

$5 i$ .

\frac{4 - 6 i}{1 + 0 - 25 i^{2}}

$\frac{4 - 6 i}{1 + 0 - 25 i^{2}}$

Step 2.3.2.10

Add

1

$1$ and

0

$0$ .

\frac{4 - 6 i}{1 - 25 i^{2}}

$\frac{4 - 6 i}{1 - 25 i^{2}}$

\frac{4 - 6 i}{1 - 25 i^{2}}

$\frac{4 - 6 i}{1 - 25 i^{2}}$

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 - 6 i}{1 - 25 \cdot - 1}

$\frac{4 - 6 i}{1 - 25 \cdot - 1}$

Step 2.3.3.2

Multiply

- 25

$- 25$ by

- 1

$- 1$ .

\frac{4 - 6 i}{1 + 25}

$\frac{4 - 6 i}{1 + 25}$

\frac{4 - 6 i}{1 + 25}

$\frac{4 - 6 i}{1 + 25}$

Step 2.3.4

Add

1

$1$ and

25

$25$ .

\frac{4 - 6 i}{26}

$\frac{4 - 6 i}{26}$

\frac{4 - 6 i}{26}

$\frac{4 - 6 i}{26}$

\frac{4 - 6 i}{26}

$\frac{4 - 6 i}{26}$

Step 3

Cancel the common factor of

4 - 6 i

$4 - 6 i$ and

26

$26$ .

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Step 3.1

Factor

2

$2$ out of

4

$4$ .

\frac{2 (2) - 6 i}{26}

$\frac{2 (2) - 6 i}{26}$

Step 3.2

Factor

2

$2$ out of

- 6 i

$- 6 i$ .

\frac{2 (2) + 2 (- 3 i)}{26}

$\frac{2 (2) + 2 (- 3 i)}{26}$

Step 3.3

Factor

2

$2$ out of

2 (2) + 2 (- 3 i)

$2 (2) + 2 (- 3 i)$ .

\frac{2 (2 - 3 i)}{26}

$\frac{2 (2 - 3 i)}{26}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor

2

$2$ out of

26

$26$ .

\frac{2 (2 - 3 i)}{2 \cdot 13}

$\frac{2 (2 - 3 i)}{2 \cdot 13}$

Step 3.4.2

Cancel the common factor.

$\frac{2 (2 - 3 i)}{2 \cdot 13}$

Step 3.4.3

Rewrite the expression.

$\frac{2 - 3 i}{13}$

Step 4

Split the fraction

$\frac{2 - 3 i}{13}$ into two fractions.

$\frac{2}{13} + \frac{- 3 i}{13}$

Step 5

Move the negative in front of the fraction.

$\frac{2}{13} - \frac{3 i}{13}$

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