Write in Standard Form (-4+3i)/(6+7i)

$\frac{- 4 + 3 i}{6 + 7 i}$

Step 1

Multiply the numerator and denominator of $\frac{- 4 + 3 i}{6 + 7 i}$ by the conjugate of $6 + 7 i$ to make the denominator real.

$\frac{- 4 + 3 i}{6 + 7 i} \cdot \frac{6 - 7 i}{6 - 7 i}$

Step 2

Multiply.

Combine.

$\frac{(- 4 + 3 i) (6 - 7 i)}{(6 + 7 i) (6 - 7 i)}$

Simplify the numerator.

Expand $(- 4 + 3 i) (6 - 7 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 4 (6 - 7 i) + 3 i (6 - 7 i)}{(6 + 7 i) (6 - 7 i)}$

Apply the distributive property.

$\frac{- 4 \cdot 6 - 4 (- 7 i) + 3 i (6 - 7 i)}{(6 + 7 i) (6 - 7 i)}$

Apply the distributive property.

$\frac{- 4 \cdot 6 - 4 (- 7 i) + 3 i \cdot 6 + 3 i (- 7 i)}{(6 + 7 i) (6 - 7 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 4$ by $6$ .

$\frac{- 24 - 4 (- 7 i) + 3 i \cdot 6 + 3 i (- 7 i)}{(6 + 7 i) (6 - 7 i)}$

Multiply $- 7$ by $- 4$ .

$\frac{- 24 + 28 i + 3 i \cdot 6 + 3 i (- 7 i)}{(6 + 7 i) (6 - 7 i)}$

Multiply $6$ by $3$ .

$\frac{- 24 + 28 i + 18 i + 3 i (- 7 i)}{(6 + 7 i) (6 - 7 i)}$

Multiply $3 i (- 7 i)$ .

Multiply $- 7$ by $3$ .

$\frac{- 24 + 28 i + 18 i - 21 i i}{(6 + 7 i) (6 - 7 i)}$

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

$\frac{- 24 + 28 i + 18 i - 21 (i^{1} i)}{(6 + 7 i) (6 - 7 i)}$

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

$\frac{- 24 + 28 i + 18 i - 21 (i^{1} i^{1})}{(6 + 7 i) (6 - 7 i)}$

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

$\frac{- 24 + 28 i + 18 i - 21 i^{1 + 1}}{(6 + 7 i) (6 - 7 i)}$

Add $1$ and $1$ .

$\frac{- 24 + 28 i + 18 i - 21 i^{2}}{(6 + 7 i) (6 - 7 i)}$

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

$\frac{- 24 + 28 i + 18 i - 21 \cdot - 1}{(6 + 7 i) (6 - 7 i)}$

Multiply $- 21$ by $- 1$ .

$\frac{- 24 + 28 i + 18 i + 21}{(6 + 7 i) (6 - 7 i)}$

Add $- 24$ and $21$ .

$\frac{- 3 + 28 i + 18 i}{(6 + 7 i) (6 - 7 i)}$

Add $28 i$ and $18 i$ .

$\frac{- 3 + 46 i}{(6 + 7 i) (6 - 7 i)}$

Simplify the denominator.

Expand $(6 + 7 i) (6 - 7 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 3 + 46 i}{6 (6 - 7 i) + 7 i (6 - 7 i)}$

Apply the distributive property.

$\frac{- 3 + 46 i}{6 \cdot 6 + 6 (- 7 i) + 7 i (6 - 7 i)}$

Apply the distributive property.

$\frac{- 3 + 46 i}{6 \cdot 6 + 6 (- 7 i) + 7 i \cdot 6 + 7 i (- 7 i)}$

Simplify.

Multiply $6$ by $6$ .

$\frac{- 3 + 46 i}{36 + 6 (- 7 i) + 7 i \cdot 6 + 7 i (- 7 i)}$

Multiply $- 7$ by $6$ .

$\frac{- 3 + 46 i}{36 - 42 i + 7 i \cdot 6 + 7 i (- 7 i)}$

Multiply $6$ by $7$ .

$\frac{- 3 + 46 i}{36 - 42 i + 42 i + 7 i (- 7 i)}$

Multiply $- 7$ by $7$ .

$\frac{- 3 + 46 i}{36 - 42 i + 42 i - 49 i i}$

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

$\frac{- 3 + 46 i}{36 - 42 i + 42 i - 49 (i^{1} i)}$

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

$\frac{- 3 + 46 i}{36 - 42 i + 42 i - 49 (i^{1} i^{1})}$

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

$\frac{- 3 + 46 i}{36 - 42 i + 42 i - 49 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 3 + 46 i}{36 - 42 i + 42 i - 49 i^{2}}$

Add $- 42 i$ and $42 i$ .

$\frac{- 3 + 46 i}{36 + 0 - 49 i^{2}}$

Add $36$ and $0$ .

$\frac{- 3 + 46 i}{36 - 49 i^{2}}$

Simplify each term.

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

$\frac{- 3 + 46 i}{36 - 49 \cdot - 1}$

Multiply $- 49$ by $- 1$ .

$\frac{- 3 + 46 i}{36 + 49}$

Add $36$ and $49$ .

$\frac{- 3 + 46 i}{85}$

Step 3

Split the fraction $\frac{- 3 + 46 i}{85}$ into two fractions.

$\frac{- 3}{85} + \frac{46 i}{85}$

Step 4

Move the negative in front of the fraction.

$- \frac{3}{85} + \frac{46 i}{85}$

Write in Standard Form (-4+3i)/(6+7i)

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