Algebra Examples

$(\frac{2 - 3 i}{1 + i}) \cdot (\frac{3 + 2 i}{3 i})$

Step 1

Multiply $\frac{2 - 3 i}{1 + i}$ by $\frac{3 + 2 i}{3 i}$ .

$\frac{(2 - 3 i) (3 + 2 i)}{(1 + i) (3 i)}$

Step 2

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

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Step 2.1

Apply the distributive property.

$\frac{2 (3 + 2 i) - 3 i (3 + 2 i)}{(1 + i) \cdot 3 i}$

Step 2.2

Apply the distributive property.

$\frac{2 \cdot 3 + 2 (2 i) - 3 i (3 + 2 i)}{(1 + i) \cdot 3 i}$

Step 2.3

Apply the distributive property.

$\frac{2 \cdot 3 + 2 (2 i) - 3 i \cdot 3 - 3 i (2 i)}{(1 + i) \cdot 3 i}$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $2$ by $3$ .

$\frac{6 + 2 (2 i) - 3 i \cdot 3 - 3 i (2 i)}{(1 + i) \cdot 3 i}$

Step 3.1.2

Multiply $2$ by $2$ .

$\frac{6 + 4 i - 3 i \cdot 3 - 3 i (2 i)}{(1 + i) \cdot 3 i}$

Step 3.1.3

Multiply $3$ by $- 3$ .

$\frac{6 + 4 i - 9 i - 3 i (2 i)}{(1 + i) \cdot 3 i}$

Step 3.1.4

Multiply $- 3 i (2 i)$ .

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Step 3.1.4.1

Multiply $2$ by $- 3$ .

$\frac{6 + 4 i - 9 i - 6 i i}{(1 + i) \cdot 3 i}$

Step 3.1.4.2

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

$\frac{6 + 4 i - 9 i - 6 (i^{1} i)}{(1 + i) \cdot 3 i}$

Step 3.1.4.3

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

$\frac{6 + 4 i - 9 i - 6 (i^{1} i^{1})}{(1 + i) \cdot 3 i}$

Step 3.1.4.4

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

$\frac{6 + 4 i - 9 i - 6 i^{1 + 1}}{(1 + i) \cdot 3 i}$

Step 3.1.4.5

Add $1$ and $1$ .

$\frac{6 + 4 i - 9 i - 6 i^{2}}{(1 + i) \cdot 3 i}$

Step 3.1.5

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

$\frac{6 + 4 i - 9 i - 6 \cdot - 1}{(1 + i) \cdot 3 i}$

Step 3.1.6

Multiply $- 6$ by $- 1$ .

$\frac{6 + 4 i - 9 i + 6}{(1 + i) \cdot 3 i}$

Step 3.2

Add $6$ and $6$ .

$\frac{12 + 4 i - 9 i}{(1 + i) \cdot 3 i}$

Step 3.3

Subtract $9 i$ from $4 i$ .

$\frac{12 - 5 i}{(1 + i) \cdot 3 i}$

Step 4

Group $i$ and $1 + i$ together.

$\frac{12 - 5 i}{i (1 + i) \cdot 3}$

Step 5

Apply the distributive property.

$\frac{12 - 5 i}{(i \cdot 1 + i i) \cdot 3}$

Step 6

Multiply $i$ by $1$ .

$\frac{12 - 5 i}{(i + i i) \cdot 3}$

Step 7

Multiply $i i$ .

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Step 7.1

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

$\frac{12 - 5 i}{(i + i^{1} i) \cdot 3}$

Step 7.2

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

$\frac{12 - 5 i}{(i + i^{1} i^{1}) \cdot 3}$

Step 7.3

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

$\frac{12 - 5 i}{(i + i^{1 + 1}) \cdot 3}$

Step 7.4

Add $1$ and $1$ .

$\frac{12 - 5 i}{(i + i^{2}) \cdot 3}$

Step 8

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

$\frac{12 - 5 i}{(i - 1) \cdot 3}$

Step 9

Simplify the expression.

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Step 9.1

Reorder $i$ and $- 1$ .

$\frac{12 - 5 i}{(- 1 + i) \cdot 3}$

Step 9.2

Move $3$ to the left of $- 1 + i$ .

$\frac{12 - 5 i}{3 \cdot (- 1 + i)}$

Step 10

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{12 - 5 i}{3 (- 1 (1) + i)}$

Step 11

Factor $- 1$ out of $i$ .

$\frac{12 - 5 i}{3 (- 1 (1) - 1 (- i))}$

Step 12

Factor $- 1$ out of $- 1 (1) - 1 (- i)$ .

$\frac{12 - 5 i}{3 (- 1 (1 - i))}$

Step 13

Move the negative in front of the fraction.

$- \frac{12 - 5 i}{3 (1 - i)}$