Write in Standard Form (7+3i)/(2-5i)

$\frac{7 + 3 i}{2 - 5 i}$

Step 1

Multiply the numerator and denominator of $\frac{7 + 3 i}{2 - 5 i}$ by the conjugate of $2 - 5 i$ to make the denominator real.

$\frac{7 + 3 i}{2 - 5 i} \cdot \frac{2 + 5 i}{2 + 5 i}$

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Expand $(7 + 3 i) (2 + 5 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $2$ .

$\frac{14 + 7 (5 i) + 3 i \cdot 2 + 3 i (5 i)}{(2 - 5 i) (2 + 5 i)}$

Multiply $5$ by $7$ .

$\frac{14 + 35 i + 3 i \cdot 2 + 3 i (5 i)}{(2 - 5 i) (2 + 5 i)}$

Multiply $2$ by $3$ .

$\frac{14 + 35 i + 6 i + 3 i (5 i)}{(2 - 5 i) (2 + 5 i)}$

Multiply $3 i (5 i)$ .

Multiply $5$ by $3$ .

$\frac{14 + 35 i + 6 i + 15 i i}{(2 - 5 i) (2 + 5 i)}$

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

$\frac{14 + 35 i + 6 i + 15 (i^{1} i)}{(2 - 5 i) (2 + 5 i)}$

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

$\frac{14 + 35 i + 6 i + 15 (i^{1} i^{1})}{(2 - 5 i) (2 + 5 i)}$

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

$\frac{14 + 35 i + 6 i + 15 i^{1 + 1}}{(2 - 5 i) (2 + 5 i)}$

Add $1$ and $1$ .

$\frac{14 + 35 i + 6 i + 15 i^{2}}{(2 - 5 i) (2 + 5 i)}$

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

$\frac{14 + 35 i + 6 i + 15 \cdot - 1}{(2 - 5 i) (2 + 5 i)}$

Multiply $15$ by $- 1$ .

$\frac{14 + 35 i + 6 i - 15}{(2 - 5 i) (2 + 5 i)}$

Subtract $15$ from $14$ .

$\frac{- 1 + 35 i + 6 i}{(2 - 5 i) (2 + 5 i)}$

Add $35 i$ and $6 i$ .

$\frac{- 1 + 41 i}{(2 - 5 i) (2 + 5 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{- 1 + 41 i}{2 (2 + 5 i) - 5 i (2 + 5 i)}$

Apply the distributive property.

$\frac{- 1 + 41 i}{2 \cdot 2 + 2 (5 i) - 5 i (2 + 5 i)}$

Apply the distributive property.

$\frac{- 1 + 41 i}{2 \cdot 2 + 2 (5 i) - 5 i \cdot 2 - 5 i (5 i)}$

Simplify.

Multiply $2$ by $2$ .

$\frac{- 1 + 41 i}{4 + 2 (5 i) - 5 i \cdot 2 - 5 i (5 i)}$

Multiply $5$ by $2$ .

$\frac{- 1 + 41 i}{4 + 10 i - 5 i \cdot 2 - 5 i (5 i)}$

Multiply $2$ by $- 5$ .

$\frac{- 1 + 41 i}{4 + 10 i - 10 i - 5 i (5 i)}$

Multiply $5$ by $- 5$ .

$\frac{- 1 + 41 i}{4 + 10 i - 10 i - 25 i i}$

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

$\frac{- 1 + 41 i}{4 + 10 i - 10 i - 25 (i^{1} i)}$

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

$\frac{- 1 + 41 i}{4 + 10 i - 10 i - 25 (i^{1} i^{1})}$

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

$\frac{- 1 + 41 i}{4 + 10 i - 10 i - 25 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 1 + 41 i}{4 + 10 i - 10 i - 25 i^{2}}$

Subtract $10 i$ from $10 i$ .

$\frac{- 1 + 41 i}{4 + 0 - 25 i^{2}}$

Add $4$ and $0$ .

$\frac{- 1 + 41 i}{4 - 25 i^{2}}$

Simplify each term.

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

$\frac{- 1 + 41 i}{4 - 25 \cdot - 1}$

Multiply $- 25$ by $- 1$ .

$\frac{- 1 + 41 i}{4 + 25}$

Add $4$ and $25$ .

$\frac{- 1 + 41 i}{29}$

Step 3

Split the fraction $\frac{- 1 + 41 i}{29}$ into two fractions.

$\frac{- 1}{29} + \frac{41 i}{29}$

Step 4

Move the negative in front of the fraction.

$- \frac{1}{29} + \frac{41 i}{29}$

Write in Standard Form (7+3i)/(2-5i)

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