Divide (10+i)/(8-2i)

$\frac{10 + i}{8 - 2 i}$

Multiply the numerator and denominator of $\frac{10 + i}{8 - 2 i}$ by the conjugate of $8 - 2 i$ to make the denominator real.

$\frac{10 + i}{8 - 2 i} \cdot \frac{8 + 2 i}{8 + 2 i}$

Multiply.

Combine.

$\frac{(10 + i) (8 + 2 i)}{(8 - 2 i) (8 + 2 i)}$

Simplify the numerator.

Expand $(10 + i) (8 + 2 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{10 (8 + 2 i) + i (8 + 2 i)}{(8 - 2 i) (8 + 2 i)}$

Apply the distributive property.

$\frac{10 \cdot 8 + 10 (2 i) + i (8 + 2 i)}{(8 - 2 i) (8 + 2 i)}$

Apply the distributive property.

$\frac{10 \cdot 8 + 10 (2 i) + i \cdot 8 + i (2 i)}{(8 - 2 i) (8 + 2 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $10$ by $8$ .

$\frac{80 + 10 (2 i) + i \cdot 8 + i (2 i)}{(8 - 2 i) (8 + 2 i)}$

Multiply $2$ by $10$ .

$\frac{80 + 20 i + i \cdot 8 + i (2 i)}{(8 - 2 i) (8 + 2 i)}$

Move $8$ to the left of $i$ .

$\frac{80 + 20 i + 8 \cdot i + i (2 i)}{(8 - 2 i) (8 + 2 i)}$

Multiply $i (2 i)$ .

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

$\frac{80 + 20 i + 8 i + 2 (i^{1} i)}{(8 - 2 i) (8 + 2 i)}$

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

$\frac{80 + 20 i + 8 i + 2 (i^{1} i^{1})}{(8 - 2 i) (8 + 2 i)}$

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

$\frac{80 + 20 i + 8 i + 2 i^{1 + 1}}{(8 - 2 i) (8 + 2 i)}$

Add $1$ and $1$ .

$\frac{80 + 20 i + 8 i + 2 i^{2}}{(8 - 2 i) (8 + 2 i)}$

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

$\frac{80 + 20 i + 8 i + 2 \cdot - 1}{(8 - 2 i) (8 + 2 i)}$

Multiply $2$ by $- 1$ .

$\frac{80 + 20 i + 8 i - 2}{(8 - 2 i) (8 + 2 i)}$

Subtract $2$ from $80$ .

$\frac{78 + 20 i + 8 i}{(8 - 2 i) (8 + 2 i)}$

Add $20 i$ and $8 i$ .

$\frac{78 + 28 i}{(8 - 2 i) (8 + 2 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{78 + 28 i}{8 (8 + 2 i) - 2 i (8 + 2 i)}$

Apply the distributive property.

$\frac{78 + 28 i}{8 \cdot 8 + 8 (2 i) - 2 i (8 + 2 i)}$

Apply the distributive property.

$\frac{78 + 28 i}{8 \cdot 8 + 8 (2 i) - 2 i \cdot 8 - 2 i (2 i)}$

Simplify.

Multiply $8$ by $8$ .

$\frac{78 + 28 i}{64 + 8 (2 i) - 2 i \cdot 8 - 2 i (2 i)}$

Multiply $2$ by $8$ .

$\frac{78 + 28 i}{64 + 16 i - 2 i \cdot 8 - 2 i (2 i)}$

Multiply $8$ by $- 2$ .

$\frac{78 + 28 i}{64 + 16 i - 16 i - 2 i (2 i)}$

Multiply $2$ by $- 2$ .

$\frac{78 + 28 i}{64 + 16 i - 16 i - 4 i i}$

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

$\frac{78 + 28 i}{64 + 16 i - 16 i - 4 (i^{1} i)}$

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

$\frac{78 + 28 i}{64 + 16 i - 16 i - 4 (i^{1} i^{1})}$

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

$\frac{78 + 28 i}{64 + 16 i - 16 i - 4 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{78 + 28 i}{64 + 16 i - 16 i - 4 i^{2}}$

Subtract $16 i$ from $16 i$ .

$\frac{78 + 28 i}{64 + 0 - 4 i^{2}}$

Add $64$ and $0$ .

$\frac{78 + 28 i}{64 - 4 i^{2}}$

Simplify each term.

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

$\frac{78 + 28 i}{64 - 4 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$\frac{78 + 28 i}{64 + 4}$

Add $64$ and $4$ .

$\frac{78 + 28 i}{68}$

Cancel the common factor of $78 + 28 i$ and $68$ .

Factor $2$ out of $78$ .

$\frac{2 (39) + 28 i}{68}$

Factor $2$ out of $28 i$ .

$\frac{2 (39) + 2 (14 i)}{68}$

Factor $2$ out of $2 (39) + 2 (14 i)$ .

$\frac{2 (39 + 14 i)}{68}$

Cancel the common factors.

Factor $2$ out of $68$ .

$\frac{2 (39 + 14 i)}{2 \cdot 34}$

Cancel the common factor.

$\frac{2 (39 + 14 i)}{2 \cdot 34}$

Rewrite the expression.

$\frac{39 + 14 i}{34}$

Split the fraction $\frac{39 + 14 i}{34}$ into two fractions.

$\frac{39}{34} + \frac{14 i}{34}$

Cancel the common factor of $14$ and $34$ .

Factor $2$ out of $14 i$ .

$\frac{39}{34} + \frac{2 (7 i)}{34}$

Cancel the common factors.

Factor $2$ out of $34$ .

$\frac{39}{34} + \frac{2 (7 i)}{2 (17)}$

Cancel the common factor.

$\frac{39}{34} + \frac{2 (7 i)}{2 \cdot 17}$

Rewrite the expression.

$\frac{39}{34} + \frac{7 i}{17}$

Divide (10+i)/(8-2i)

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