Simplify 5/(2+i)

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

Step 1

Multiply the numerator and denominator of $\frac{5}{2 + 1 i}$ by the conjugate of $2 + 1 i$ to make the denominator real.

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $5$ by $2$ .

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

Multiply $- 1$ by $5$ .

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

Simplify the denominator.

Expand $(2 + 1 i) (2 - i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $2$ by $2$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $2$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Add $- 2 i$ and $2 i$ .

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

Add $4$ and $0$ .

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

Simplify each term.

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

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

Multiply $- 1$ by $- 1$ .

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

Add $4$ and $1$ .

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

Step 3

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

Factor $5$ out of $10$ .

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

Factor $5$ out of $- 5 i$ .

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

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

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

Cancel the common factors.

Factor $5$ out of $5$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2 - i$ by $1$ .

$2 - i$

Simplify 5/(2+i)

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