Simplify (2+i)/(2-i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

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

Move $2$ to the left of $i$ .

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

Multiply $i i$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Subtract $1$ from $4$ .

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

Add $2 i$ and $2 i$ .

$\frac{3 + 4 i}{(2 - i) (2 + i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{3 + 4 i}{2 (2 + i) - i (2 + i)}$

Apply the distributive property.

$\frac{3 + 4 i}{2 \cdot 2 + 2 i - i (2 + i)}$

Apply the distributive property.

$\frac{3 + 4 i}{2 \cdot 2 + 2 i - i \cdot 2 - i i}$

Simplify.

Multiply $2$ by $2$ .

$\frac{3 + 4 i}{4 + 2 i - i \cdot 2 - i i}$

Multiply $2$ by $- 1$ .

$\frac{3 + 4 i}{4 + 2 i - 2 i - i i}$

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

$\frac{3 + 4 i}{4 + 2 i - 2 i - (i^{1} i)}$

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

$\frac{3 + 4 i}{4 + 2 i - 2 i - (i^{1} i^{1})}$

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

$\frac{3 + 4 i}{4 + 2 i - 2 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{3 + 4 i}{4 + 2 i - 2 i - i^{2}}$

Subtract $2 i$ from $2 i$ .

$\frac{3 + 4 i}{4 + 0 - i^{2}}$

Add $4$ and $0$ .

$\frac{3 + 4 i}{4 - i^{2}}$

Simplify each term.

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

$\frac{3 + 4 i}{4 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{3 + 4 i}{4 + 1}$

Add $4$ and $1$ .

$\frac{3 + 4 i}{5}$

Step 3

Split the fraction $\frac{3 + 4 i}{5}$ into two fractions.

$\frac{3}{5} + \frac{4 i}{5}$

Simplify (2+i)/(2-i)

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