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

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

Multiply $i (- i)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 1$ by $- 1$ .

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

Multiply $2$ by $- 2$ .

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

Multiply $- 1$ by $2$ .

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

Subtract $2 i$ from $- 2 i$ .

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

Subtract $4$ from $1$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $- 2$ by $- 2$ .

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

Multiply $- 1$ by $- 2$ .

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

Multiply $- 2$ by $1$ .

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

Multiply $- 1$ 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.

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

Step 4

Simplify each term.

Move the negative in front of the fraction.

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

Move the negative in front of the fraction.

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

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

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