Algebra Examples

$\frac{6}{5 + i}$

Step 1

Multiply the numerator and denominator of $\frac{6}{5 + 1 i}$ by the conjugate of $5 + 1 i$ to make the denominator real.

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Multiply $6$ by $5$ .

$\frac{30 + 6 (- i)}{(5 + 1 i) (5 - i)}$

Step 2.2.3

Multiply $- 1$ by $6$ .

$\frac{30 - 6 i}{(5 + 1 i) (5 - i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{30 - 6 i}{5 (5 - i) + 1 i (5 - i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{30 - 6 i}{5 \cdot 5 + 5 (- i) + 1 i (5 - i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{30 - 6 i}{5 \cdot 5 + 5 (- i) + 1 i \cdot 5 + 1 i (- i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $5$ by $5$ .

$\frac{30 - 6 i}{25 + 5 (- i) + 1 i \cdot 5 + 1 i (- i)}$

Step 2.3.2.2

Multiply $- 1$ by $5$ .

$\frac{30 - 6 i}{25 - 5 i + 1 i \cdot 5 + 1 i (- i)}$

Step 2.3.2.3

Multiply $5$ by $1$ .

$\frac{30 - 6 i}{25 - 5 i + 5 i + 1 i (- i)}$

Step 2.3.2.4

Multiply $- 1$ by $1$ .

$\frac{30 - 6 i}{25 - 5 i + 5 i - i i}$

Step 2.3.2.5

Raise $i$ to the power of $1$ .

$\frac{30 - 6 i}{25 - 5 i + 5 i - (i^{1} i)}$

Step 2.3.2.6

Raise $i$ to the power of $1$ .

$\frac{30 - 6 i}{25 - 5 i + 5 i - (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{30 - 6 i}{25 - 5 i + 5 i - i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{30 - 6 i}{25 - 5 i + 5 i - i^{2}}$

Step 2.3.2.9

Add $- 5 i$ and $5 i$ .

$\frac{30 - 6 i}{25 + 0 - i^{2}}$

Step 2.3.2.10

Add $25$ and $0$ .

$\frac{30 - 6 i}{25 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite $i^{2}$ as $- 1$ .

$\frac{30 - 6 i}{25 - - 1}$

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{30 - 6 i}{25 + 1}$

Step 2.3.4

Add $25$ and $1$ .

$\frac{30 - 6 i}{26}$

Step 3

Cancel the common factor of $30 - 6 i$ and $26$ .

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Step 3.1

Factor $2$ out of $30$ .

$\frac{2 (15) - 6 i}{26}$

Step 3.2

Factor $2$ out of $- 6 i$ .

$\frac{2 (15) + 2 (- 3 i)}{26}$

Step 3.3

Factor $2$ out of $2 (15) + 2 (- 3 i)$ .

$\frac{2 (15 - 3 i)}{26}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $2$ out of $26$ .

$\frac{2 (15 - 3 i)}{2 \cdot 13}$

Step 3.4.2

Cancel the common factor.

$\frac{2 (15 - 3 i)}{2 \cdot 13}$

Step 3.4.3

Rewrite the expression.

$\frac{15 - 3 i}{13}$

Step 4

Split the fraction $\frac{15 - 3 i}{13}$ into two fractions.

$\frac{15}{13} + \frac{- 3 i}{13}$

Step 5

Move the negative in front of the fraction.

$\frac{15}{13} - \frac{3 i}{13}$