Simplify (-46+54i)/(-5-7i)

$\frac{- 46 + 54 i}{- 5 - 7 i}$

Step 1

Multiply the numerator and denominator of $\frac{- 46 + 54 i}{- 5 - 7 i}$ by the conjugate of $- 5 - 7 i$ to make the denominator real.

$\frac{- 46 + 54 i}{- 5 - 7 i} \cdot \frac{- 5 + 7 i}{- 5 + 7 i}$

Step 2

Multiply.

Combine.

$\frac{(- 46 + 54 i) (- 5 + 7 i)}{(- 5 - 7 i) (- 5 + 7 i)}$

Simplify the numerator.

Expand $(- 46 + 54 i) (- 5 + 7 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 46 (- 5 + 7 i) + 54 i (- 5 + 7 i)}{(- 5 - 7 i) (- 5 + 7 i)}$

Apply the distributive property.

$\frac{- 46 \cdot - 5 - 46 (7 i) + 54 i (- 5 + 7 i)}{(- 5 - 7 i) (- 5 + 7 i)}$

Apply the distributive property.

$\frac{- 46 \cdot - 5 - 46 (7 i) + 54 i \cdot - 5 + 54 i (7 i)}{(- 5 - 7 i) (- 5 + 7 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 46$ by $- 5$ .

$\frac{230 - 46 (7 i) + 54 i \cdot - 5 + 54 i (7 i)}{(- 5 - 7 i) (- 5 + 7 i)}$

Multiply $7$ by $- 46$ .

$\frac{230 - 322 i + 54 i \cdot - 5 + 54 i (7 i)}{(- 5 - 7 i) (- 5 + 7 i)}$

Multiply $- 5$ by $54$ .

$\frac{230 - 322 i - 270 i + 54 i (7 i)}{(- 5 - 7 i) (- 5 + 7 i)}$

Multiply $54 i (7 i)$ .

Multiply $7$ by $54$ .

$\frac{230 - 322 i - 270 i + 378 i i}{(- 5 - 7 i) (- 5 + 7 i)}$

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

$\frac{230 - 322 i - 270 i + 378 (i^{1} i)}{(- 5 - 7 i) (- 5 + 7 i)}$

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

$\frac{230 - 322 i - 270 i + 378 (i^{1} i^{1})}{(- 5 - 7 i) (- 5 + 7 i)}$

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

$\frac{230 - 322 i - 270 i + 378 i^{1 + 1}}{(- 5 - 7 i) (- 5 + 7 i)}$

Add $1$ and $1$ .

$\frac{230 - 322 i - 270 i + 378 i^{2}}{(- 5 - 7 i) (- 5 + 7 i)}$

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

$\frac{230 - 322 i - 270 i + 378 \cdot - 1}{(- 5 - 7 i) (- 5 + 7 i)}$

Multiply $378$ by $- 1$ .

$\frac{230 - 322 i - 270 i - 378}{(- 5 - 7 i) (- 5 + 7 i)}$

Subtract $378$ from $230$ .

$\frac{- 148 - 322 i - 270 i}{(- 5 - 7 i) (- 5 + 7 i)}$

Subtract $270 i$ from $- 322 i$ .

$\frac{- 148 - 592 i}{(- 5 - 7 i) (- 5 + 7 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{- 148 - 592 i}{- 5 (- 5 + 7 i) - 7 i (- 5 + 7 i)}$

Apply the distributive property.

$\frac{- 148 - 592 i}{- 5 \cdot - 5 - 5 (7 i) - 7 i (- 5 + 7 i)}$

Apply the distributive property.

$\frac{- 148 - 592 i}{- 5 \cdot - 5 - 5 (7 i) - 7 i \cdot - 5 - 7 i (7 i)}$

Simplify.

Multiply $- 5$ by $- 5$ .

$\frac{- 148 - 592 i}{25 - 5 (7 i) - 7 i \cdot - 5 - 7 i (7 i)}$

Multiply $7$ by $- 5$ .

$\frac{- 148 - 592 i}{25 - 35 i - 7 i \cdot - 5 - 7 i (7 i)}$

Multiply $- 5$ by $- 7$ .

$\frac{- 148 - 592 i}{25 - 35 i + 35 i - 7 i (7 i)}$

Multiply $7$ by $- 7$ .

$\frac{- 148 - 592 i}{25 - 35 i + 35 i - 49 i i}$

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

$\frac{- 148 - 592 i}{25 - 35 i + 35 i - 49 (i^{1} i)}$

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

$\frac{- 148 - 592 i}{25 - 35 i + 35 i - 49 (i^{1} i^{1})}$

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

$\frac{- 148 - 592 i}{25 - 35 i + 35 i - 49 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 148 - 592 i}{25 - 35 i + 35 i - 49 i^{2}}$

Add $- 35 i$ and $35 i$ .

$\frac{- 148 - 592 i}{25 + 0 - 49 i^{2}}$

Add $25$ and $0$ .

$\frac{- 148 - 592 i}{25 - 49 i^{2}}$

Simplify each term.

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

$\frac{- 148 - 592 i}{25 - 49 \cdot - 1}$

Multiply $- 49$ by $- 1$ .

$\frac{- 148 - 592 i}{25 + 49}$

Add $25$ and $49$ .

$\frac{- 148 - 592 i}{74}$

Step 3

Cancel the common factor of $- 148 - 592 i$ and $74$ .

Factor $74$ out of $- 148$ .

$\frac{74 \cdot - 2 - 592 i}{74}$

Factor $74$ out of $- 592 i$ .

$\frac{74 \cdot - 2 + 74 (- 8 i)}{74}$

Factor $74$ out of $74 (- 2) + 74 (- 8 i)$ .

$\frac{74 (- 2 - 8 i)}{74}$

Cancel the common factors.

Factor $74$ out of $74$ .

$\frac{74 (- 2 - 8 i)}{74 (1)}$

Cancel the common factor.

$\frac{74 (- 2 - 8 i)}{74 \cdot 1}$

Rewrite the expression.

$\frac{- 2 - 8 i}{1}$

Divide $- 2 - 8 i$ by $1$ .

$- 2 - 8 i$

Simplify (-46+54i)/(-5-7i)

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