Precalculus Examples

$\frac{3}{2 + 4 i}$

Step 1

Multiply the numerator and denominator of $\frac{3}{2 + 4 i}$ by the conjugate of $2 + 4 i$ to make the denominator real.

$\frac{3}{2 + 4 i} \cdot \frac{2 - 4 i}{2 - 4 i}$

Step 2

Multiply.

Tap for more steps...

Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

Tap for more steps...

Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Multiply $3$ by $2$ .

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

Step 2.2.3

Multiply $- 4$ by $3$ .

$\frac{6 - 12 i}{(2 + 4 i) (2 - 4 i)}$

Step 2.3

Simplify the denominator.

Tap for more steps...

Step 2.3.1

Expand $(2 + 4 i) (2 - 4 i)$ using the FOIL Method.

Tap for more steps...

Step 2.3.1.1

Apply the distributive property.

$\frac{6 - 12 i}{2 (2 - 4 i) + 4 i (2 - 4 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{6 - 12 i}{2 \cdot 2 + 2 (- 4 i) + 4 i (2 - 4 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{6 - 12 i}{2 \cdot 2 + 2 (- 4 i) + 4 i \cdot 2 + 4 i (- 4 i)}$

Step 2.3.2

Simplify.

Tap for more steps...

Step 2.3.2.1

Multiply $2$ by $2$ .

$\frac{6 - 12 i}{4 + 2 (- 4 i) + 4 i \cdot 2 + 4 i (- 4 i)}$

Step 2.3.2.2

Multiply $- 4$ by $2$ .

$\frac{6 - 12 i}{4 - 8 i + 4 i \cdot 2 + 4 i (- 4 i)}$

Step 2.3.2.3

Multiply $2$ by $4$ .

$\frac{6 - 12 i}{4 - 8 i + 8 i + 4 i (- 4 i)}$

Step 2.3.2.4

Multiply $- 4$ by $4$ .

$\frac{6 - 12 i}{4 - 8 i + 8 i - 16 i i}$

Step 2.3.2.5

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

$\frac{6 - 12 i}{4 - 8 i + 8 i - 16 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{6 - 12 i}{4 - 8 i + 8 i - 16 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{6 - 12 i}{4 - 8 i + 8 i - 16 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{6 - 12 i}{4 - 8 i + 8 i - 16 i^{2}}$

Step 2.3.2.9

Add $- 8 i$ and $8 i$ .

$\frac{6 - 12 i}{4 + 0 - 16 i^{2}}$

Step 2.3.2.10

Add $4$ and $0$ .

$\frac{6 - 12 i}{4 - 16 i^{2}}$

Step 2.3.3

Simplify each term.

Tap for more steps...

Step 2.3.3.1

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

$\frac{6 - 12 i}{4 - 16 \cdot - 1}$

Step 2.3.3.2

Multiply $- 16$ by $- 1$ .

$\frac{6 - 12 i}{4 + 16}$

Step 2.3.4

Add $4$ and $16$ .

$\frac{6 - 12 i}{20}$

Step 3

Cancel the common factor of $6 - 12 i$ and $20$ .

Tap for more steps...

Step 3.1

Factor $2$ out of $6$ .

$\frac{2 (3) - 12 i}{20}$

Step 3.2

Factor $2$ out of $- 12 i$ .

$\frac{2 (3) + 2 (- 6 i)}{20}$

Step 3.3

Factor $2$ out of $2 (3) + 2 (- 6 i)$ .

$\frac{2 (3 - 6 i)}{20}$

Step 3.4

Cancel the common factors.

Tap for more steps...

Step 3.4.1

Factor $2$ out of $20$ .

$\frac{2 (3 - 6 i)}{2 \cdot 10}$

Step 3.4.2

Cancel the common factor.

$\frac{2 (3 - 6 i)}{2 \cdot 10}$

Step 3.4.3

Rewrite the expression.

$\frac{3 - 6 i}{10}$

Step 4

Split the fraction $\frac{3 - 6 i}{10}$ into two fractions.

$\frac{3}{10} + \frac{- 6 i}{10}$

Step 5

Simplify each term.

Tap for more steps...

Step 5.1

Cancel the common factor of $- 6$ and $10$ .

Tap for more steps...

Step 5.1.1

Factor $2$ out of $- 6 i$ .

$\frac{3}{10} + \frac{2 (- 3 i)}{10}$

Step 5.1.2

Cancel the common factors.

Tap for more steps...

Step 5.1.2.1

Factor $2$ out of $10$ .

$\frac{3}{10} + \frac{2 (- 3 i)}{2 (5)}$

Step 5.1.2.2

Cancel the common factor.

$\frac{3}{10} + \frac{2 (- 3 i)}{2 \cdot 5}$

Step 5.1.2.3

Rewrite the expression.

$\frac{3}{10} + \frac{- 3 i}{5}$

Step 5.2

Move the negative in front of the fraction.

$\frac{3}{10} - \frac{3 i}{5}$