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

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

Step 1

Cancel the common factor of $8 + 2 i$ and $8 - 4 i$ .

Factor $2$ out of $8$ .

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

Factor $2$ out of $2 i$ .

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

Factor $2$ out of $2 \cdot 4 + 2 (i)$ .

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

Cancel the common factors.

Factor $2$ out of $8$ .

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

Factor $2$ out of $- 4 i$ .

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

Factor $2$ out of $2 (4) + 2 (- 2 i)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

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

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

Step 3

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

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

Multiply $2$ by $4$ .

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

Move $4$ to the left of $i$ .

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

Multiply $i (2 i)$ .

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

$\frac{16 + 8 i + 4 i + 2 (i^{1} i)}{(4 - 2 i) (4 + 2 i)}$

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

$\frac{16 + 8 i + 4 i + 2 (i^{1} i^{1})}{(4 - 2 i) (4 + 2 i)}$

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

$\frac{16 + 8 i + 4 i + 2 i^{1 + 1}}{(4 - 2 i) (4 + 2 i)}$

Add $1$ and $1$ .

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

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

$\frac{16 + 8 i + 4 i + 2 \cdot - 1}{(4 - 2 i) (4 + 2 i)}$

Multiply $2$ by $- 1$ .

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

Subtract $2$ from $16$ .

$\frac{14 + 8 i + 4 i}{(4 - 2 i) (4 + 2 i)}$

Add $8 i$ and $4 i$ .

$\frac{14 + 12 i}{(4 - 2 i) (4 + 2 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{14 + 12 i}{4 (4 + 2 i) - 2 i (4 + 2 i)}$

Apply the distributive property.

$\frac{14 + 12 i}{4 \cdot 4 + 4 (2 i) - 2 i (4 + 2 i)}$

Apply the distributive property.

$\frac{14 + 12 i}{4 \cdot 4 + 4 (2 i) - 2 i \cdot 4 - 2 i (2 i)}$

Simplify.

Multiply $4$ by $4$ .

$\frac{14 + 12 i}{16 + 4 (2 i) - 2 i \cdot 4 - 2 i (2 i)}$

Multiply $2$ by $4$ .

$\frac{14 + 12 i}{16 + 8 i - 2 i \cdot 4 - 2 i (2 i)}$

Multiply $4$ by $- 2$ .

$\frac{14 + 12 i}{16 + 8 i - 8 i - 2 i (2 i)}$

Multiply $2$ by $- 2$ .

$\frac{14 + 12 i}{16 + 8 i - 8 i - 4 i i}$

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

$\frac{14 + 12 i}{16 + 8 i - 8 i - 4 (i^{1} i)}$

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

$\frac{14 + 12 i}{16 + 8 i - 8 i - 4 (i^{1} i^{1})}$

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

$\frac{14 + 12 i}{16 + 8 i - 8 i - 4 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{14 + 12 i}{16 + 8 i - 8 i - 4 i^{2}}$

Subtract $8 i$ from $8 i$ .

$\frac{14 + 12 i}{16 + 0 - 4 i^{2}}$

Add $16$ and $0$ .

$\frac{14 + 12 i}{16 - 4 i^{2}}$

Simplify each term.

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

$\frac{14 + 12 i}{16 - 4 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$\frac{14 + 12 i}{16 + 4}$

Add $16$ and $4$ .

$\frac{14 + 12 i}{20}$

Step 4

Cancel the common factor of $14 + 12 i$ and $20$ .

Factor $2$ out of $14$ .

$\frac{2 (7) + 12 i}{20}$

Factor $2$ out of $12 i$ .

$\frac{2 (7) + 2 (6 i)}{20}$

Factor $2$ out of $2 (7) + 2 (6 i)$ .

$\frac{2 (7 + 6 i)}{20}$

Cancel the common factors.

Factor $2$ out of $20$ .

$\frac{2 (7 + 6 i)}{2 \cdot 10}$

Cancel the common factor.

$\frac{2 (7 + 6 i)}{2 \cdot 10}$

Rewrite the expression.

$\frac{7 + 6 i}{10}$

Step 5

Split the fraction $\frac{7 + 6 i}{10}$ into two fractions.

$\frac{7}{10} + \frac{6 i}{10}$

Step 6

Cancel the common factor of $6$ and $10$ .

Factor $2$ out of $6 i$ .

$\frac{7}{10} + \frac{2 (3 i)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$\frac{7}{10} + \frac{2 (3 i)}{2 (5)}$

Cancel the common factor.

$\frac{7}{10} + \frac{2 (3 i)}{2 \cdot 5}$

Rewrite the expression.

$\frac{7}{10} + \frac{3 i}{5}$

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

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