Precalculus Examples

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

Step 1

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

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

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

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

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 $5$ by $5$ .

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

Step 2.2.2.1.2

Move $5$ to the left of $i$ .

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

Step 2.2.2.1.3

Multiply $i i$ .

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Step 2.2.2.1.3.1

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

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

Step 2.2.2.1.3.2

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

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

Step 2.2.2.1.3.3

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

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

Step 2.2.2.1.3.4

Add $1$ and $1$ .

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

Step 2.2.2.1.4

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

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

Step 2.2.2.2

Subtract $1$ from $25$ .

$\frac{24 + 5 i + 5 i}{(5 - i) (5 + i)}$

Step 2.2.2.3

Add $5 i$ and $5 i$ .

$\frac{24 + 10 i}{(5 - i) (5 + i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{24 + 10 i}{5 (5 + i) - i (5 + i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{24 + 10 i}{5 \cdot 5 + 5 i - i (5 + i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{24 + 10 i}{5 \cdot 5 + 5 i - i \cdot 5 - i i}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $5$ by $5$ .

$\frac{24 + 10 i}{25 + 5 i - i \cdot 5 - i i}$

Step 2.3.2.2

Multiply $5$ by $- 1$ .

$\frac{24 + 10 i}{25 + 5 i - 5 i - i i}$

Step 2.3.2.3

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

$\frac{24 + 10 i}{25 + 5 i - 5 i - (i^{1} i)}$

Step 2.3.2.4

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

$\frac{24 + 10 i}{25 + 5 i - 5 i - (i^{1} i^{1})}$

Step 2.3.2.5

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

$\frac{24 + 10 i}{25 + 5 i - 5 i - i^{1 + 1}}$

Step 2.3.2.6

Add $1$ and $1$ .

$\frac{24 + 10 i}{25 + 5 i - 5 i - i^{2}}$

Step 2.3.2.7

Subtract $5 i$ from $5 i$ .

$\frac{24 + 10 i}{25 + 0 - i^{2}}$

Step 2.3.2.8

Add $25$ and $0$ .

$\frac{24 + 10 i}{25 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{24 + 10 i}{25 - - 1}$

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{24 + 10 i}{25 + 1}$

Step 2.3.4

Add $25$ and $1$ .

$\frac{24 + 10 i}{26}$

Step 3

Cancel the common factor of $24 + 10 i$ and $26$ .

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Step 3.1

Factor $2$ out of $24$ .

$\frac{2 (12) + 10 i}{26}$

Step 3.2

Factor $2$ out of $10 i$ .

$\frac{2 (12) + 2 (5 i)}{26}$

Step 3.3

Factor $2$ out of $2 (12) + 2 (5 i)$ .

$\frac{2 (12 + 5 i)}{26}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $2$ out of $26$ .

$\frac{2 (12 + 5 i)}{2 \cdot 13}$

Step 3.4.2

Cancel the common factor.

$\frac{2 (12 + 5 i)}{2 \cdot 13}$

Step 3.4.3

Rewrite the expression.

$\frac{12 + 5 i}{13}$

Step 4

Split the fraction $\frac{12 + 5 i}{13}$ into two fractions.

$\frac{12}{13} + \frac{5 i}{13}$