Algebra Examples

\frac{2}{5 + i}

$\frac{2}{5 + i}$

Step 1

Multiply the numerator and denominator of

\frac{2}{5 + 1 i}

$\frac{2}{5 + 1 i}$ by the conjugate of

5 + 1 i

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

\frac{2}{5 + 1 i} \cdot \frac{5 - i}{5 - i}

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

Step 2

Multiply.

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Step 2.1

Combine.

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

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

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

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

Step 2.2.2

Multiply

2

$2$ by

5

$5$ .

\frac{10 + 2 (- i)}{(5 + 1 i) (5 - i)}

$\frac{10 + 2 (- i)}{(5 + 1 i) (5 - i)}$

Step 2.2.3

Multiply

- 1

$- 1$ by

2

$2$ .

\frac{10 - 2 i}{(5 + 1 i) (5 - i)}

$\frac{10 - 2 i}{(5 + 1 i) (5 - i)}$

\frac{10 - 2 i}{(5 + 1 i) (5 - i)}

$\frac{10 - 2 i}{(5 + 1 i) (5 - i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

(5 + 1 i) (5 - i)

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

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Step 2.3.1.1

Apply the distributive property.

\frac{10 - 2 i}{5 (5 - i) + 1 i (5 - i)}

$\frac{10 - 2 i}{5 (5 - i) + 1 i (5 - i)}$

Step 2.3.1.2

Apply the distributive property.

\frac{10 - 2 i}{5 \cdot 5 + 5 (- i) + 1 i (5 - i)}

$\frac{10 - 2 i}{5 \cdot 5 + 5 (- i) + 1 i (5 - i)}$

Step 2.3.1.3

Apply the distributive property.

\frac{10 - 2 i}{5 \cdot 5 + 5 (- i) + 1 i \cdot 5 + 1 i (- i)}

$\frac{10 - 2 i}{5 \cdot 5 + 5 (- i) + 1 i \cdot 5 + 1 i (- i)}$

\frac{10 - 2 i}{5 \cdot 5 + 5 (- i) + 1 i \cdot 5 + 1 i (- i)}

$\frac{10 - 2 i}{5 \cdot 5 + 5 (- i) + 1 i \cdot 5 + 1 i (- i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

5

$5$ by

5

$5$ .

\frac{10 - 2 i}{25 + 5 (- i) + 1 i \cdot 5 + 1 i (- i)}

$\frac{10 - 2 i}{25 + 5 (- i) + 1 i \cdot 5 + 1 i (- i)}$

Step 2.3.2.2

Multiply

- 1

$- 1$ by

5

$5$ .

\frac{10 - 2 i}{25 - 5 i + 1 i \cdot 5 + 1 i (- i)}

$\frac{10 - 2 i}{25 - 5 i + 1 i \cdot 5 + 1 i (- i)}$

Step 2.3.2.3

Multiply

5

$5$ by

1

$1$ .

\frac{10 - 2 i}{25 - 5 i + 5 i + 1 i (- i)}

$\frac{10 - 2 i}{25 - 5 i + 5 i + 1 i (- i)}$

Step 2.3.2.4

Multiply

- 1

$- 1$ by

1

$1$ .

\frac{10 - 2 i}{25 - 5 i + 5 i - i i}

$\frac{10 - 2 i}{25 - 5 i + 5 i - i i}$

Step 2.3.2.5

Raise

i

$i$ to the power of

1

$1$ .

\frac{10 - 2 i}{25 - 5 i + 5 i - (i^{1} i)}

$\frac{10 - 2 i}{25 - 5 i + 5 i - (i^{1} i)}$

Step 2.3.2.6

Raise

i

$i$ to the power of

1

$1$ .

\frac{10 - 2 i}{25 - 5 i + 5 i - (i^{1} i^{1})}

$\frac{10 - 2 i}{25 - 5 i + 5 i - (i^{1} i^{1})}$

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{10 - 2 i}{25 - 5 i + 5 i - i^{1 + 1}}

$\frac{10 - 2 i}{25 - 5 i + 5 i - i^{1 + 1}}$

Step 2.3.2.8

Add

1

$1$ and

1

$1$ .

\frac{10 - 2 i}{25 - 5 i + 5 i - i^{2}}

$\frac{10 - 2 i}{25 - 5 i + 5 i - i^{2}}$

Step 2.3.2.9

Add

- 5 i

$- 5 i$ and

5 i

$5 i$ .

\frac{10 - 2 i}{25 + 0 - i^{2}}

$\frac{10 - 2 i}{25 + 0 - i^{2}}$

Step 2.3.2.10

Add

25

$25$ and

0

$0$ .

\frac{10 - 2 i}{25 - i^{2}}

$\frac{10 - 2 i}{25 - i^{2}}$

\frac{10 - 2 i}{25 - i^{2}}

$\frac{10 - 2 i}{25 - i^{2}}$

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{10 - 2 i}{25 - - 1}

$\frac{10 - 2 i}{25 - - 1}$

Step 2.3.3.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{10 - 2 i}{25 + 1}

$\frac{10 - 2 i}{25 + 1}$

\frac{10 - 2 i}{25 + 1}

$\frac{10 - 2 i}{25 + 1}$

Step 2.3.4

Add

25

$25$ and

1

$1$ .

\frac{10 - 2 i}{26}

$\frac{10 - 2 i}{26}$

\frac{10 - 2 i}{26}

$\frac{10 - 2 i}{26}$

\frac{10 - 2 i}{26}

$\frac{10 - 2 i}{26}$

Step 3

Cancel the common factor of

10 - 2 i

$10 - 2 i$ and

26

$26$ .

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Step 3.1

Factor

2

$2$ out of

10

$10$ .

\frac{2 (5) - 2 i}{26}

$\frac{2 (5) - 2 i}{26}$

Step 3.2

Factor

2

$2$ out of

- 2 i

$- 2 i$ .

\frac{2 (5) + 2 (- i)}{26}

$\frac{2 (5) + 2 (- i)}{26}$

Step 3.3

Factor

2

$2$ out of

2 (5) + 2 (- i)

$2 (5) + 2 (- i)$ .

\frac{2 (5 - i)}{26}

$\frac{2 (5 - i)}{26}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor

2

$2$ out of

26

$26$ .

\frac{2 (5 - i)}{2 \cdot 13}

$\frac{2 (5 - i)}{2 \cdot 13}$

Step 3.4.2

Cancel the common factor.

$\frac{2 (5 - i)}{2 \cdot 13}$

Step 3.4.3

Rewrite the expression.

$\frac{5 - i}{13}$

Step 4

Split the fraction

$\frac{5 - i}{13}$ into two fractions.

$\frac{5}{13} + \frac{- i}{13}$

Step 5

Move the negative in front of the fraction.

$\frac{5}{13} - \frac{i}{13}$