Write in Standard Form (9i)/(4+4i)

$\frac{9 i}{4 + 4 i}$

Step 1

Multiply the numerator and denominator of $\frac{9 i}{4 + 4 i}$ by the conjugate of $4 + 4 i$ to make the denominator real.

$\frac{9 i}{4 + 4 i} \cdot \frac{4 - 4 i}{4 - 4 i}$

Step 2

Multiply.

Combine.

$\frac{9 i (4 - 4 i)}{(4 + 4 i) (4 - 4 i)}$

Simplify the numerator.

Apply the distributive property.

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

Multiply $4$ by $9$ .

$\frac{36 i + 9 i (- 4 i)}{(4 + 4 i) (4 - 4 i)}$

Multiply $9 i (- 4 i)$ .

Multiply $- 4$ by $9$ .

$\frac{36 i - 36 i i}{(4 + 4 i) (4 - 4 i)}$

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

$\frac{36 i - 36 (i^{1} i)}{(4 + 4 i) (4 - 4 i)}$

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

$\frac{36 i - 36 (i^{1} i^{1})}{(4 + 4 i) (4 - 4 i)}$

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

$\frac{36 i - 36 i^{1 + 1}}{(4 + 4 i) (4 - 4 i)}$

Add $1$ and $1$ .

$\frac{36 i - 36 i^{2}}{(4 + 4 i) (4 - 4 i)}$

Simplify each term.

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

$\frac{36 i - 36 \cdot - 1}{(4 + 4 i) (4 - 4 i)}$

Multiply $- 36$ by $- 1$ .

$\frac{36 i + 36}{(4 + 4 i) (4 - 4 i)}$

Reorder $36 i$ and $36$ .

$\frac{36 + 36 i}{(4 + 4 i) (4 - 4 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{36 + 36 i}{4 (4 - 4 i) + 4 i (4 - 4 i)}$

Apply the distributive property.

$\frac{36 + 36 i}{4 \cdot 4 + 4 (- 4 i) + 4 i (4 - 4 i)}$

Apply the distributive property.

$\frac{36 + 36 i}{4 \cdot 4 + 4 (- 4 i) + 4 i \cdot 4 + 4 i (- 4 i)}$

Simplify.

Multiply $4$ by $4$ .

$\frac{36 + 36 i}{16 + 4 (- 4 i) + 4 i \cdot 4 + 4 i (- 4 i)}$

Multiply $- 4$ by $4$ .

$\frac{36 + 36 i}{16 - 16 i + 4 i \cdot 4 + 4 i (- 4 i)}$

Multiply $4$ by $4$ .

$\frac{36 + 36 i}{16 - 16 i + 16 i + 4 i (- 4 i)}$

Multiply $- 4$ by $4$ .

$\frac{36 + 36 i}{16 - 16 i + 16 i - 16 i i}$

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

$\frac{36 + 36 i}{16 - 16 i + 16 i - 16 (i^{1} i)}$

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

$\frac{36 + 36 i}{16 - 16 i + 16 i - 16 (i^{1} i^{1})}$

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

$\frac{36 + 36 i}{16 - 16 i + 16 i - 16 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{36 + 36 i}{16 - 16 i + 16 i - 16 i^{2}}$

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

$\frac{36 + 36 i}{16 + 0 - 16 i^{2}}$

Add $16$ and $0$ .

$\frac{36 + 36 i}{16 - 16 i^{2}}$

Simplify each term.

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

$\frac{36 + 36 i}{16 - 16 \cdot - 1}$

Multiply $- 16$ by $- 1$ .

$\frac{36 + 36 i}{16 + 16}$

Add $16$ and $16$ .

$\frac{36 + 36 i}{32}$

Step 3

Cancel the common factor of $36 + 36 i$ and $32$ .

Factor $4$ out of $36$ .

$\frac{4 (9) + 36 i}{32}$

Factor $4$ out of $36 i$ .

$\frac{4 (9) + 4 (9 i)}{32}$

Factor $4$ out of $4 (9) + 4 (9 i)$ .

$\frac{4 (9 + 9 i)}{32}$

Cancel the common factors.

Factor $4$ out of $32$ .

$\frac{4 (9 + 9 i)}{4 \cdot 8}$

Cancel the common factor.

$\frac{4 (9 + 9 i)}{4 \cdot 8}$

Rewrite the expression.

$\frac{9 + 9 i}{8}$

Step 4

Split the fraction $\frac{9 + 9 i}{8}$ into two fractions.

$\frac{9}{8} + \frac{9 i}{8}$

Write in Standard Form (9i)/(4+4i)

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