Write in Standard Form (10-15i)/(1-2i)

$\frac{10 - 15 i}{1 - 2 i}$

Step 1

Multiply the numerator and denominator of $\frac{10 - 15 i}{1 - 2 i}$ by the conjugate of $1 - 2 i$ to make the denominator real.

$\frac{10 - 15 i}{1 - 2 i} \cdot \frac{1 + 2 i}{1 + 2 i}$

Step 2

Multiply.

Combine.

$\frac{(10 - 15 i) (1 + 2 i)}{(1 - 2 i) (1 + 2 i)}$

Simplify the numerator.

Expand $(10 - 15 i) (1 + 2 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{10 (1 + 2 i) - 15 i (1 + 2 i)}{(1 - 2 i) (1 + 2 i)}$

Apply the distributive property.

$\frac{10 \cdot 1 + 10 (2 i) - 15 i (1 + 2 i)}{(1 - 2 i) (1 + 2 i)}$

Apply the distributive property.

$\frac{10 \cdot 1 + 10 (2 i) - 15 i \cdot 1 - 15 i (2 i)}{(1 - 2 i) (1 + 2 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $10$ by $1$ .

$\frac{10 + 10 (2 i) - 15 i \cdot 1 - 15 i (2 i)}{(1 - 2 i) (1 + 2 i)}$

Multiply $2$ by $10$ .

$\frac{10 + 20 i - 15 i \cdot 1 - 15 i (2 i)}{(1 - 2 i) (1 + 2 i)}$

Multiply $- 15$ by $1$ .

$\frac{10 + 20 i - 15 i - 15 i (2 i)}{(1 - 2 i) (1 + 2 i)}$

Multiply $- 15 i (2 i)$ .

Multiply $2$ by $- 15$ .

$\frac{10 + 20 i - 15 i - 30 i i}{(1 - 2 i) (1 + 2 i)}$

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

$\frac{10 + 20 i - 15 i - 30 (i^{1} i)}{(1 - 2 i) (1 + 2 i)}$

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

$\frac{10 + 20 i - 15 i - 30 (i^{1} i^{1})}{(1 - 2 i) (1 + 2 i)}$

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

$\frac{10 + 20 i - 15 i - 30 i^{1 + 1}}{(1 - 2 i) (1 + 2 i)}$

Add $1$ and $1$ .

$\frac{10 + 20 i - 15 i - 30 i^{2}}{(1 - 2 i) (1 + 2 i)}$

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

$\frac{10 + 20 i - 15 i - 30 \cdot - 1}{(1 - 2 i) (1 + 2 i)}$

Multiply $- 30$ by $- 1$ .

$\frac{10 + 20 i - 15 i + 30}{(1 - 2 i) (1 + 2 i)}$

Add $10$ and $30$ .

$\frac{40 + 20 i - 15 i}{(1 - 2 i) (1 + 2 i)}$

Subtract $15 i$ from $20 i$ .

$\frac{40 + 5 i}{(1 - 2 i) (1 + 2 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{40 + 5 i}{1 (1 + 2 i) - 2 i (1 + 2 i)}$

Apply the distributive property.

$\frac{40 + 5 i}{1 \cdot 1 + 1 (2 i) - 2 i (1 + 2 i)}$

Apply the distributive property.

$\frac{40 + 5 i}{1 \cdot 1 + 1 (2 i) - 2 i \cdot 1 - 2 i (2 i)}$

Simplify.

Multiply $1$ by $1$ .

$\frac{40 + 5 i}{1 + 1 (2 i) - 2 i \cdot 1 - 2 i (2 i)}$

Multiply $2$ by $1$ .

$\frac{40 + 5 i}{1 + 2 i - 2 i \cdot 1 - 2 i (2 i)}$

Multiply $- 2$ by $1$ .

$\frac{40 + 5 i}{1 + 2 i - 2 i - 2 i (2 i)}$

Multiply $2$ by $- 2$ .

$\frac{40 + 5 i}{1 + 2 i - 2 i - 4 i i}$

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

$\frac{40 + 5 i}{1 + 2 i - 2 i - 4 (i^{1} i)}$

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

$\frac{40 + 5 i}{1 + 2 i - 2 i - 4 (i^{1} i^{1})}$

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

$\frac{40 + 5 i}{1 + 2 i - 2 i - 4 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{40 + 5 i}{1 + 2 i - 2 i - 4 i^{2}}$

Subtract $2 i$ from $2 i$ .

$\frac{40 + 5 i}{1 + 0 - 4 i^{2}}$

Add $1$ and $0$ .

$\frac{40 + 5 i}{1 - 4 i^{2}}$

Simplify each term.

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

$\frac{40 + 5 i}{1 - 4 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$\frac{40 + 5 i}{1 + 4}$

Add $1$ and $4$ .

$\frac{40 + 5 i}{5}$

Step 3

Cancel the common factor of $40 + 5 i$ and $5$ .

Factor $5$ out of $40$ .

$\frac{5 \cdot 8 + 5 i}{5}$

Factor $5$ out of $5 i$ .

$\frac{5 \cdot 8 + 5 (i)}{5}$

Factor $5$ out of $5 (8) + 5 (i)$ .

$\frac{5 (8 + i)}{5}$

Cancel the common factors.

Factor $5$ out of $5$ .

$\frac{5 (8 + i)}{5 (1)}$

Cancel the common factor.

$\frac{5 (8 + i)}{5 \cdot 1}$

Rewrite the expression.

$\frac{8 + i}{1}$

Divide $8 + i$ by $1$ .

$8 + i$

Write in Standard Form (10-15i)/(1-2i)

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