Precalculus Examples

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

Step 1

Multiply the numerator and denominator of $\frac{5}{2 - i}$ by the conjugate of $2 - i$ to make the denominator real.

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Multiply $5$ by $2$ .

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(2 - i) (2 + i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

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

Step 2.3.1.2

Apply the distributive property.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $2$ by $2$ .

$\frac{10 + 5 i}{4 + 2 i - i \cdot 2 - i i}$

Step 2.3.2.2

Multiply $2$ by $- 1$ .

$\frac{10 + 5 i}{4 + 2 i - 2 i - i i}$

Step 2.3.2.3

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

$\frac{10 + 5 i}{4 + 2 i - 2 i - (i^{1} i)}$

Step 2.3.2.4

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

$\frac{10 + 5 i}{4 + 2 i - 2 i - (i^{1} i^{1})}$

Step 2.3.2.5

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

$\frac{10 + 5 i}{4 + 2 i - 2 i - i^{1 + 1}}$

Step 2.3.2.6

Add $1$ and $1$ .

$\frac{10 + 5 i}{4 + 2 i - 2 i - i^{2}}$

Step 2.3.2.7

Subtract $2 i$ from $2 i$ .

$\frac{10 + 5 i}{4 + 0 - i^{2}}$

Step 2.3.2.8

Add $4$ and $0$ .

$\frac{10 + 5 i}{4 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{10 + 5 i}{4 - - 1}$

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{10 + 5 i}{4 + 1}$

Step 2.3.4

Add $4$ and $1$ .

$\frac{10 + 5 i}{5}$

Step 3

Cancel the common factor of $10 + 5 i$ and $5$ .

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Step 3.1

Factor $5$ out of $10$ .

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

Step 3.2

Factor $5$ out of $5 i$ .

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

Step 3.3

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

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

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $5$ out of $5$ .

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 3.4.4

Divide $2 + i$ by $1$ .

$2 + i$

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