Simplify (24+2i)/(-3+7i)

$\frac{24 + 2 i}{- 3 + 7 i}$

Step 1

Multiply the numerator and denominator of $\frac{24 + 2 i}{- 3 + 7 i}$ by the conjugate of $- 3 + 7 i$ to make the denominator real.

$\frac{24 + 2 i}{- 3 + 7 i} \cdot \frac{- 3 - 7 i}{- 3 - 7 i}$

Step 2

Multiply.

Combine.

$\frac{(24 + 2 i) (- 3 - 7 i)}{(- 3 + 7 i) (- 3 - 7 i)}$

Simplify the numerator.

Expand $(24 + 2 i) (- 3 - 7 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{24 (- 3 - 7 i) + 2 i (- 3 - 7 i)}{(- 3 + 7 i) (- 3 - 7 i)}$

Apply the distributive property.

$\frac{24 \cdot - 3 + 24 (- 7 i) + 2 i (- 3 - 7 i)}{(- 3 + 7 i) (- 3 - 7 i)}$

Apply the distributive property.

$\frac{24 \cdot - 3 + 24 (- 7 i) + 2 i \cdot - 3 + 2 i (- 7 i)}{(- 3 + 7 i) (- 3 - 7 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $24$ by $- 3$ .

$\frac{- 72 + 24 (- 7 i) + 2 i \cdot - 3 + 2 i (- 7 i)}{(- 3 + 7 i) (- 3 - 7 i)}$

Multiply $- 7$ by $24$ .

$\frac{- 72 - 168 i + 2 i \cdot - 3 + 2 i (- 7 i)}{(- 3 + 7 i) (- 3 - 7 i)}$

Multiply $- 3$ by $2$ .

$\frac{- 72 - 168 i - 6 i + 2 i (- 7 i)}{(- 3 + 7 i) (- 3 - 7 i)}$

Multiply $2 i (- 7 i)$ .

Multiply $- 7$ by $2$ .

$\frac{- 72 - 168 i - 6 i - 14 i i}{(- 3 + 7 i) (- 3 - 7 i)}$

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

$\frac{- 72 - 168 i - 6 i - 14 (i^{1} i)}{(- 3 + 7 i) (- 3 - 7 i)}$

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

$\frac{- 72 - 168 i - 6 i - 14 (i^{1} i^{1})}{(- 3 + 7 i) (- 3 - 7 i)}$

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

$\frac{- 72 - 168 i - 6 i - 14 i^{1 + 1}}{(- 3 + 7 i) (- 3 - 7 i)}$

Add $1$ and $1$ .

$\frac{- 72 - 168 i - 6 i - 14 i^{2}}{(- 3 + 7 i) (- 3 - 7 i)}$

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

$\frac{- 72 - 168 i - 6 i - 14 \cdot - 1}{(- 3 + 7 i) (- 3 - 7 i)}$

Multiply $- 14$ by $- 1$ .

$\frac{- 72 - 168 i - 6 i + 14}{(- 3 + 7 i) (- 3 - 7 i)}$

Add $- 72$ and $14$ .

$\frac{- 58 - 168 i - 6 i}{(- 3 + 7 i) (- 3 - 7 i)}$

Subtract $6 i$ from $- 168 i$ .

$\frac{- 58 - 174 i}{(- 3 + 7 i) (- 3 - 7 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{- 58 - 174 i}{- 3 (- 3 - 7 i) + 7 i (- 3 - 7 i)}$

Apply the distributive property.

$\frac{- 58 - 174 i}{- 3 \cdot - 3 - 3 (- 7 i) + 7 i (- 3 - 7 i)}$

Apply the distributive property.

$\frac{- 58 - 174 i}{- 3 \cdot - 3 - 3 (- 7 i) + 7 i \cdot - 3 + 7 i (- 7 i)}$

Simplify.

Multiply $- 3$ by $- 3$ .

$\frac{- 58 - 174 i}{9 - 3 (- 7 i) + 7 i \cdot - 3 + 7 i (- 7 i)}$

Multiply $- 7$ by $- 3$ .

$\frac{- 58 - 174 i}{9 + 21 i + 7 i \cdot - 3 + 7 i (- 7 i)}$

Multiply $- 3$ by $7$ .

$\frac{- 58 - 174 i}{9 + 21 i - 21 i + 7 i (- 7 i)}$

Multiply $- 7$ by $7$ .

$\frac{- 58 - 174 i}{9 + 21 i - 21 i - 49 i i}$

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

$\frac{- 58 - 174 i}{9 + 21 i - 21 i - 49 (i^{1} i)}$

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

$\frac{- 58 - 174 i}{9 + 21 i - 21 i - 49 (i^{1} i^{1})}$

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

$\frac{- 58 - 174 i}{9 + 21 i - 21 i - 49 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 58 - 174 i}{9 + 21 i - 21 i - 49 i^{2}}$

Subtract $21 i$ from $21 i$ .

$\frac{- 58 - 174 i}{9 + 0 - 49 i^{2}}$

Add $9$ and $0$ .

$\frac{- 58 - 174 i}{9 - 49 i^{2}}$

Simplify each term.

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

$\frac{- 58 - 174 i}{9 - 49 \cdot - 1}$

Multiply $- 49$ by $- 1$ .

$\frac{- 58 - 174 i}{9 + 49}$

Add $9$ and $49$ .

$\frac{- 58 - 174 i}{58}$

Step 3

Cancel the common factor of $- 58 - 174 i$ and $58$ .

Factor $58$ out of $- 58$ .

$\frac{58 \cdot - 1 - 174 i}{58}$

Factor $58$ out of $- 174 i$ .

$\frac{58 \cdot - 1 + 58 (- 3 i)}{58}$

Factor $58$ out of $58 (- 1) + 58 (- 3 i)$ .

$\frac{58 (- 1 - 3 i)}{58}$

Cancel the common factors.

Factor $58$ out of $58$ .

$\frac{58 (- 1 - 3 i)}{58 (1)}$

Cancel the common factor.

$\frac{58 (- 1 - 3 i)}{58 \cdot 1}$

Rewrite the expression.

$\frac{- 1 - 3 i}{1}$

Divide $- 1 - 3 i$ by $1$ .

$- 1 - 3 i$

Simplify (24+2i)/(-3+7i)

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