Write in Standard Form (1-3i)/(2+5i)

$\frac{1 - 3 i}{2 + 5 i}$

Step 1

Multiply the numerator and denominator of $\frac{1 - 3 i}{2 + 5 i}$ by the conjugate of $2 + 5 i$ to make the denominator real.

$\frac{1 - 3 i}{2 + 5 i} \cdot \frac{2 - 5 i}{2 - 5 i}$

Step 2

Multiply.

Combine.

$\frac{(1 - 3 i) (2 - 5 i)}{(2 + 5 i) (2 - 5 i)}$

Simplify the numerator.

Expand $(1 - 3 i) (2 - 5 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{1 (2 - 5 i) - 3 i (2 - 5 i)}{(2 + 5 i) (2 - 5 i)}$

Apply the distributive property.

$\frac{1 \cdot 2 + 1 (- 5 i) - 3 i (2 - 5 i)}{(2 + 5 i) (2 - 5 i)}$

Apply the distributive property.

$\frac{1 \cdot 2 + 1 (- 5 i) - 3 i \cdot 2 - 3 i (- 5 i)}{(2 + 5 i) (2 - 5 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $1$ .

$\frac{2 + 1 (- 5 i) - 3 i \cdot 2 - 3 i (- 5 i)}{(2 + 5 i) (2 - 5 i)}$

Multiply $- 5 i$ by $1$ .

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

Multiply $2$ by $- 3$ .

$\frac{2 - 5 i - 6 i - 3 i (- 5 i)}{(2 + 5 i) (2 - 5 i)}$

Multiply $- 3 i (- 5 i)$ .

Multiply $- 5$ by $- 3$ .

$\frac{2 - 5 i - 6 i + 15 i i}{(2 + 5 i) (2 - 5 i)}$

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

$\frac{2 - 5 i - 6 i + 15 (i^{1} i)}{(2 + 5 i) (2 - 5 i)}$

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

$\frac{2 - 5 i - 6 i + 15 (i^{1} i^{1})}{(2 + 5 i) (2 - 5 i)}$

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

$\frac{2 - 5 i - 6 i + 15 i^{1 + 1}}{(2 + 5 i) (2 - 5 i)}$

Add $1$ and $1$ .

$\frac{2 - 5 i - 6 i + 15 i^{2}}{(2 + 5 i) (2 - 5 i)}$

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

$\frac{2 - 5 i - 6 i + 15 \cdot - 1}{(2 + 5 i) (2 - 5 i)}$

Multiply $15$ by $- 1$ .

$\frac{2 - 5 i - 6 i - 15}{(2 + 5 i) (2 - 5 i)}$

Subtract $15$ from $2$ .

$\frac{- 13 - 5 i - 6 i}{(2 + 5 i) (2 - 5 i)}$

Subtract $6 i$ from $- 5 i$ .

$\frac{- 13 - 11 i}{(2 + 5 i) (2 - 5 i)}$

Simplify the denominator.

Expand $(2 + 5 i) (2 - 5 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 13 - 11 i}{2 (2 - 5 i) + 5 i (2 - 5 i)}$

Apply the distributive property.

$\frac{- 13 - 11 i}{2 \cdot 2 + 2 (- 5 i) + 5 i (2 - 5 i)}$

Apply the distributive property.

$\frac{- 13 - 11 i}{2 \cdot 2 + 2 (- 5 i) + 5 i \cdot 2 + 5 i (- 5 i)}$

Simplify.

Multiply $2$ by $2$ .

$\frac{- 13 - 11 i}{4 + 2 (- 5 i) + 5 i \cdot 2 + 5 i (- 5 i)}$

Multiply $- 5$ by $2$ .

$\frac{- 13 - 11 i}{4 - 10 i + 5 i \cdot 2 + 5 i (- 5 i)}$

Multiply $2$ by $5$ .

$\frac{- 13 - 11 i}{4 - 10 i + 10 i + 5 i (- 5 i)}$

Multiply $- 5$ by $5$ .

$\frac{- 13 - 11 i}{4 - 10 i + 10 i - 25 i i}$

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

$\frac{- 13 - 11 i}{4 - 10 i + 10 i - 25 (i^{1} i)}$

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

$\frac{- 13 - 11 i}{4 - 10 i + 10 i - 25 (i^{1} i^{1})}$

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

$\frac{- 13 - 11 i}{4 - 10 i + 10 i - 25 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 13 - 11 i}{4 - 10 i + 10 i - 25 i^{2}}$

Add $- 10 i$ and $10 i$ .

$\frac{- 13 - 11 i}{4 + 0 - 25 i^{2}}$

Add $4$ and $0$ .

$\frac{- 13 - 11 i}{4 - 25 i^{2}}$

Simplify each term.

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

$\frac{- 13 - 11 i}{4 - 25 \cdot - 1}$

Multiply $- 25$ by $- 1$ .

$\frac{- 13 - 11 i}{4 + 25}$

Add $4$ and $25$ .

$\frac{- 13 - 11 i}{29}$

Step 3

Split the fraction $\frac{- 13 - 11 i}{29}$ into two fractions.

$\frac{- 13}{29} + \frac{- 11 i}{29}$

Step 4

Simplify each term.

Move the negative in front of the fraction.

$- \frac{13}{29} + \frac{- 11 i}{29}$

Move the negative in front of the fraction.

$- \frac{13}{29} - \frac{11 i}{29}$

Write in Standard Form (1-3i)/(2+5i)

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