Simplify (1+8i)/(2-4i)

$\frac{1 + 8 i}{2 - 4 i}$

Multiply the numerator and denominator of $\frac{1 + 8 i}{2 - 4 i}$ by the conjugate of $2 - 4 i$ to make the denominator real.

$\frac{1 + 8 i}{2 - 4 i} \cdot \frac{2 + 4 i}{2 + 4 i}$

Multiply.

Combine.

$\frac{(1 + 8 i) (2 + 4 i)}{(2 - 4 i) (2 + 4 i)}$

Simplify the numerator.

Expand $(1 + 8 i) (2 + 4 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{1 (2 + 4 i) + 8 i (2 + 4 i)}{(2 - 4 i) (2 + 4 i)}$

Apply the distributive property.

$\frac{1 \cdot 2 + 1 (4 i) + 8 i (2 + 4 i)}{(2 - 4 i) (2 + 4 i)}$

Apply the distributive property.

$\frac{1 \cdot 2 + 1 (4 i) + 8 i \cdot 2 + 8 i (4 i)}{(2 - 4 i) (2 + 4 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $1$ .

$\frac{2 + 1 (4 i) + 8 i \cdot 2 + 8 i (4 i)}{(2 - 4 i) (2 + 4 i)}$

Multiply $4 i$ by $1$ .

$\frac{2 + 4 i + 8 i \cdot 2 + 8 i (4 i)}{(2 - 4 i) (2 + 4 i)}$

Multiply $2$ by $8$ .

$\frac{2 + 4 i + 16 i + 8 i (4 i)}{(2 - 4 i) (2 + 4 i)}$

Multiply $8 i (4 i)$ .

Multiply $4$ by $8$ .

$\frac{2 + 4 i + 16 i + 32 i i}{(2 - 4 i) (2 + 4 i)}$

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

$\frac{2 + 4 i + 16 i + 32 (i^{1} i)}{(2 - 4 i) (2 + 4 i)}$

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

$\frac{2 + 4 i + 16 i + 32 (i^{1} i^{1})}{(2 - 4 i) (2 + 4 i)}$

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

$\frac{2 + 4 i + 16 i + 32 i^{1 + 1}}{(2 - 4 i) (2 + 4 i)}$

Add $1$ and $1$ .

$\frac{2 + 4 i + 16 i + 32 i^{2}}{(2 - 4 i) (2 + 4 i)}$

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

$\frac{2 + 4 i + 16 i + 32 \cdot - 1}{(2 - 4 i) (2 + 4 i)}$

Multiply $32$ by $- 1$ .

$\frac{2 + 4 i + 16 i - 32}{(2 - 4 i) (2 + 4 i)}$

Subtract $32$ from $2$ .

$\frac{- 30 + 4 i + 16 i}{(2 - 4 i) (2 + 4 i)}$

Add $4 i$ and $16 i$ .

$\frac{- 30 + 20 i}{(2 - 4 i) (2 + 4 i)}$

Simplify the denominator.

Expand $(2 - 4 i) (2 + 4 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 30 + 20 i}{2 (2 + 4 i) - 4 i (2 + 4 i)}$

Apply the distributive property.

$\frac{- 30 + 20 i}{2 \cdot 2 + 2 (4 i) - 4 i (2 + 4 i)}$

Apply the distributive property.

$\frac{- 30 + 20 i}{2 \cdot 2 + 2 (4 i) - 4 i \cdot 2 - 4 i (4 i)}$

Simplify.

Multiply $2$ by $2$ .

$\frac{- 30 + 20 i}{4 + 2 (4 i) - 4 i \cdot 2 - 4 i (4 i)}$

Multiply $4$ by $2$ .

$\frac{- 30 + 20 i}{4 + 8 i - 4 i \cdot 2 - 4 i (4 i)}$

Multiply $2$ by $- 4$ .

$\frac{- 30 + 20 i}{4 + 8 i - 8 i - 4 i (4 i)}$

Multiply $4$ by $- 4$ .

$\frac{- 30 + 20 i}{4 + 8 i - 8 i - 16 i i}$

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

$\frac{- 30 + 20 i}{4 + 8 i - 8 i - 16 (i^{1} i)}$

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

$\frac{- 30 + 20 i}{4 + 8 i - 8 i - 16 (i^{1} i^{1})}$

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

$\frac{- 30 + 20 i}{4 + 8 i - 8 i - 16 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 30 + 20 i}{4 + 8 i - 8 i - 16 i^{2}}$

Subtract $8 i$ from $8 i$ .

$\frac{- 30 + 20 i}{4 + 0 - 16 i^{2}}$

Add $4$ and $0$ .

$\frac{- 30 + 20 i}{4 - 16 i^{2}}$

Simplify each term.

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

$\frac{- 30 + 20 i}{4 - 16 \cdot - 1}$

Multiply $- 16$ by $- 1$ .

$\frac{- 30 + 20 i}{4 + 16}$

Add $4$ and $16$ .

$\frac{- 30 + 20 i}{20}$

Cancel the common factor of $- 30 + 20 i$ and $20$ .

Factor $10$ out of $- 30$ .

$\frac{10 (- 3) + 20 i}{20}$

Factor $10$ out of $20 i$ .

$\frac{10 (- 3) + 10 (2 i)}{20}$

Factor $10$ out of $10 (- 3) + 10 (2 i)$ .

$\frac{10 (- 3 + 2 i)}{20}$

Cancel the common factors.

Factor $10$ out of $20$ .

$\frac{10 (- 3 + 2 i)}{10 \cdot 2}$

Cancel the common factor.

$\frac{10 (- 3 + 2 i)}{10 \cdot 2}$

Rewrite the expression.

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

Split the fraction $\frac{- 3 + 2 i}{2}$ into two fractions.

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

Simplify each term.

Move the negative in front of the fraction.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $i$ by $1$ .

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

Simplify (1+8i)/(2-4i)

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