Simplify (-4+i)/(6-7i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- 4$ by $6$ .

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

Multiply $7$ by $- 4$ .

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

Move $6$ to the left of $i$ .

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

Multiply $i (7 i)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $7$ by $- 1$ .

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

Subtract $7$ from $- 24$ .

$\frac{- 31 - 28 i + 6 i}{(6 - 7 i) (6 + 7 i)}$

Add $- 28 i$ and $6 i$ .

$\frac{- 31 - 22 i}{(6 - 7 i) (6 + 7 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{- 31 - 22 i}{6 (6 + 7 i) - 7 i (6 + 7 i)}$

Apply the distributive property.

$\frac{- 31 - 22 i}{6 \cdot 6 + 6 (7 i) - 7 i (6 + 7 i)}$

Apply the distributive property.

$\frac{- 31 - 22 i}{6 \cdot 6 + 6 (7 i) - 7 i \cdot 6 - 7 i (7 i)}$

Simplify.

Multiply $6$ by $6$ .

$\frac{- 31 - 22 i}{36 + 6 (7 i) - 7 i \cdot 6 - 7 i (7 i)}$

Multiply $7$ by $6$ .

$\frac{- 31 - 22 i}{36 + 42 i - 7 i \cdot 6 - 7 i (7 i)}$

Multiply $6$ by $- 7$ .

$\frac{- 31 - 22 i}{36 + 42 i - 42 i - 7 i (7 i)}$

Multiply $7$ by $- 7$ .

$\frac{- 31 - 22 i}{36 + 42 i - 42 i - 49 i i}$

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

$\frac{- 31 - 22 i}{36 + 42 i - 42 i - 49 (i^{1} i)}$

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

$\frac{- 31 - 22 i}{36 + 42 i - 42 i - 49 (i^{1} i^{1})}$

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

$\frac{- 31 - 22 i}{36 + 42 i - 42 i - 49 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 31 - 22 i}{36 + 42 i - 42 i - 49 i^{2}}$

Subtract $42 i$ from $42 i$ .

$\frac{- 31 - 22 i}{36 + 0 - 49 i^{2}}$

Add $36$ and $0$ .

$\frac{- 31 - 22 i}{36 - 49 i^{2}}$

Simplify each term.

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

$\frac{- 31 - 22 i}{36 - 49 \cdot - 1}$

Multiply $- 49$ by $- 1$ .

$\frac{- 31 - 22 i}{36 + 49}$

Add $36$ and $49$ .

$\frac{- 31 - 22 i}{85}$

Step 3

Split the fraction $\frac{- 31 - 22 i}{85}$ into two fractions.

$\frac{- 31}{85} + \frac{- 22 i}{85}$

Step 4

Simplify each term.

Move the negative in front of the fraction.

$- \frac{31}{85} + \frac{- 22 i}{85}$

Move the negative in front of the fraction.

$- \frac{31}{85} - \frac{22 i}{85}$

Simplify (-4+i)/(6-7i)

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