Algebra Examples

\frac{2}{5 - i}

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

Step 1

Multiply the numerator and denominator of

\frac{2}{5 - i}

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

5 - i

$5 - i$ to make the denominator real.

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

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

Step 2

Multiply.

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Step 2.1

Combine.

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

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

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 - i) (5 + i)}

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

Step 2.2.2

Multiply

2

$2$ by

5

$5$ .

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

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

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

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

(5 - i) (5 + i)

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

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Step 2.3.1.1

Apply the distributive property.

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

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

Step 2.3.1.2

Apply the distributive property.

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

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

Step 2.3.1.3

Apply the distributive property.

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

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

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

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

5

$5$ by

5

$5$ .

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

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

Step 2.3.2.2

Multiply

5

$5$ 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 2.3.2.3

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 2.3.2.4

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 2.3.2.5

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 2.3.2.6

Add

$1$ and

$1$ .

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

Step 2.3.2.7

Subtract

$5 i$ from

$5 i$ .

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

Step 2.3.2.8

Add

$25$ and

$0$ .

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

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

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

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

Step 2.3.3.2

Multiply

$- 1$ by

$- 1$ .

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

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

Step 2.3.4

Add

$25$ and

$1$ .

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

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

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

Step 3

Cancel the common factor of

$10 + 2 i$ and

$26$ .

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Step 3.1

Factor

$2$ out of

$10$ .

$\frac{2 \cdot 5 + 2 i}{26}$

Step 3.2

Factor

$2$ out of

$2 i$ .

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

Step 3.3

Factor

$2$ out of

$2 \cdot 5 + 2 (i)$ .

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

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor

$2$ out of

$26$ .

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

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

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

Step 4

Split the fraction

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

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

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

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$