Write in Standard Form (3i)/(4+4i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $4$ by $3$ .

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

Multiply $3 i (- 4 i)$ .

Multiply $- 4$ by $3$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify each term.

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

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

Multiply $- 12$ by $- 1$ .

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

Reorder $12 i$ and $12$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $4$ by $4$ .

$\frac{12 + 12 i}{16 + 4 (- 4 i) + 4 i \cdot 4 + 4 i (- 4 i)}$

Multiply $- 4$ by $4$ .

$\frac{12 + 12 i}{16 - 16 i + 4 i \cdot 4 + 4 i (- 4 i)}$

Multiply $4$ by $4$ .

$\frac{12 + 12 i}{16 - 16 i + 16 i + 4 i (- 4 i)}$

Multiply $- 4$ by $4$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Add $- 16 i$ and $16 i$ .

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

Add $16$ and $0$ .

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

Simplify each term.

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

$\frac{12 + 12 i}{16 - 16 \cdot - 1}$

Multiply $- 16$ by $- 1$ .

$\frac{12 + 12 i}{16 + 16}$

Add $16$ and $16$ .

$\frac{12 + 12 i}{32}$

Step 3

Cancel the common factor of $12 + 12 i$ and $32$ .

Factor $4$ out of $12$ .

$\frac{4 (3) + 12 i}{32}$

Factor $4$ out of $12 i$ .

$\frac{4 (3) + 4 (3 i)}{32}$

Factor $4$ out of $4 (3) + 4 (3 i)$ .

$\frac{4 (3 + 3 i)}{32}$

Cancel the common factors.

Factor $4$ out of $32$ .

$\frac{4 (3 + 3 i)}{4 \cdot 8}$

Cancel the common factor.

$\frac{4 (3 + 3 i)}{4 \cdot 8}$

Rewrite the expression.

$\frac{3 + 3 i}{8}$

Step 4

Split the fraction $\frac{3 + 3 i}{8}$ into two fractions.

$\frac{3}{8} + \frac{3 i}{8}$

Write in Standard Form (3i)/(4+4i)

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