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

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $3$ .

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

Multiply $3$ by $2$ .

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

Multiply $3$ by $7$ .

$\frac{6 + 6 i + 21 i + 7 i (3 i)}{(3 - 3 i) (3 + 3 i)}$

Multiply $7 i (3 i)$ .

Multiply $3$ by $7$ .

$\frac{6 + 6 i + 21 i + 21 i i}{(3 - 3 i) (3 + 3 i)}$

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

$\frac{6 + 6 i + 21 i + 21 (i^{1} i)}{(3 - 3 i) (3 + 3 i)}$

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

$\frac{6 + 6 i + 21 i + 21 (i^{1} i^{1})}{(3 - 3 i) (3 + 3 i)}$

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

$\frac{6 + 6 i + 21 i + 21 i^{1 + 1}}{(3 - 3 i) (3 + 3 i)}$

Add $1$ and $1$ .

$\frac{6 + 6 i + 21 i + 21 i^{2}}{(3 - 3 i) (3 + 3 i)}$

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

$\frac{6 + 6 i + 21 i + 21 \cdot - 1}{(3 - 3 i) (3 + 3 i)}$

Multiply $21$ by $- 1$ .

$\frac{6 + 6 i + 21 i - 21}{(3 - 3 i) (3 + 3 i)}$

Subtract $21$ from $6$ .

$\frac{- 15 + 6 i + 21 i}{(3 - 3 i) (3 + 3 i)}$

Add $6 i$ and $21 i$ .

$\frac{- 15 + 27 i}{(3 - 3 i) (3 + 3 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{- 15 + 27 i}{3 (3 + 3 i) - 3 i (3 + 3 i)}$

Apply the distributive property.

$\frac{- 15 + 27 i}{3 \cdot 3 + 3 (3 i) - 3 i (3 + 3 i)}$

Apply the distributive property.

$\frac{- 15 + 27 i}{3 \cdot 3 + 3 (3 i) - 3 i \cdot 3 - 3 i (3 i)}$

Simplify.

Multiply $3$ by $3$ .

$\frac{- 15 + 27 i}{9 + 3 (3 i) - 3 i \cdot 3 - 3 i (3 i)}$

Multiply $3$ by $3$ .

$\frac{- 15 + 27 i}{9 + 9 i - 3 i \cdot 3 - 3 i (3 i)}$

Multiply $3$ by $- 3$ .

$\frac{- 15 + 27 i}{9 + 9 i - 9 i - 3 i (3 i)}$

Multiply $3$ by $- 3$ .

$\frac{- 15 + 27 i}{9 + 9 i - 9 i - 9 i i}$

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

$\frac{- 15 + 27 i}{9 + 9 i - 9 i - 9 (i^{1} i)}$

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

$\frac{- 15 + 27 i}{9 + 9 i - 9 i - 9 (i^{1} i^{1})}$

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

$\frac{- 15 + 27 i}{9 + 9 i - 9 i - 9 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 15 + 27 i}{9 + 9 i - 9 i - 9 i^{2}}$

Subtract $9 i$ from $9 i$ .

$\frac{- 15 + 27 i}{9 + 0 - 9 i^{2}}$

Add $9$ and $0$ .

$\frac{- 15 + 27 i}{9 - 9 i^{2}}$

Simplify each term.

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

$\frac{- 15 + 27 i}{9 - 9 \cdot - 1}$

Multiply $- 9$ by $- 1$ .

$\frac{- 15 + 27 i}{9 + 9}$

Add $9$ and $9$ .

$\frac{- 15 + 27 i}{18}$

Step 3

Cancel the common factor of $- 15 + 27 i$ and $18$ .

Factor $3$ out of $- 15$ .

$\frac{3 (- 5) + 27 i}{18}$

Factor $3$ out of $27 i$ .

$\frac{3 (- 5) + 3 (9 i)}{18}$

Factor $3$ out of $3 (- 5) + 3 (9 i)$ .

$\frac{3 (- 5 + 9 i)}{18}$

Cancel the common factors.

Factor $3$ out of $18$ .

$\frac{3 (- 5 + 9 i)}{3 \cdot 6}$

Cancel the common factor.

$\frac{3 (- 5 + 9 i)}{3 \cdot 6}$

Rewrite the expression.

$\frac{- 5 + 9 i}{6}$

Step 4

Split the fraction $\frac{- 5 + 9 i}{6}$ into two fractions.

$\frac{- 5}{6} + \frac{9 i}{6}$

Step 5

Simplify each term.

Move the negative in front of the fraction.

$- \frac{5}{6} + \frac{9 i}{6}$

Cancel the common factor of $9$ and $6$ .

Factor $3$ out of $9 i$ .

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

Cancel the common factors.

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

$- \frac{5}{6} + \frac{3 i}{2}$

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

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