Write in Standard Form (5i)/(6+6i)

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

Step 1

Multiply the numerator and denominator of $\frac{5 i}{6 + 6 i}$ by the conjugate of $6 + 6 i$ to make the denominator real.

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $6$ by $5$ .

$\frac{30 i + 5 i (- 6 i)}{(6 + 6 i) (6 - 6 i)}$

Multiply $5 i (- 6 i)$ .

Multiply $- 6$ by $5$ .

$\frac{30 i - 30 i i}{(6 + 6 i) (6 - 6 i)}$

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

$\frac{30 i - 30 (i^{1} i)}{(6 + 6 i) (6 - 6 i)}$

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

$\frac{30 i - 30 (i^{1} i^{1})}{(6 + 6 i) (6 - 6 i)}$

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

$\frac{30 i - 30 i^{1 + 1}}{(6 + 6 i) (6 - 6 i)}$

Add $1$ and $1$ .

$\frac{30 i - 30 i^{2}}{(6 + 6 i) (6 - 6 i)}$

Simplify each term.

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

$\frac{30 i - 30 \cdot - 1}{(6 + 6 i) (6 - 6 i)}$

Multiply $- 30$ by $- 1$ .

$\frac{30 i + 30}{(6 + 6 i) (6 - 6 i)}$

Reorder $30 i$ and $30$ .

$\frac{30 + 30 i}{(6 + 6 i) (6 - 6 i)}$

Simplify the denominator.

Expand $(6 + 6 i) (6 - 6 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{30 + 30 i}{6 (6 - 6 i) + 6 i (6 - 6 i)}$

Apply the distributive property.

$\frac{30 + 30 i}{6 \cdot 6 + 6 (- 6 i) + 6 i (6 - 6 i)}$

Apply the distributive property.

$\frac{30 + 30 i}{6 \cdot 6 + 6 (- 6 i) + 6 i \cdot 6 + 6 i (- 6 i)}$

Simplify.

Multiply $6$ by $6$ .

$\frac{30 + 30 i}{36 + 6 (- 6 i) + 6 i \cdot 6 + 6 i (- 6 i)}$

Multiply $- 6$ by $6$ .

$\frac{30 + 30 i}{36 - 36 i + 6 i \cdot 6 + 6 i (- 6 i)}$

Multiply $6$ by $6$ .

$\frac{30 + 30 i}{36 - 36 i + 36 i + 6 i (- 6 i)}$

Multiply $- 6$ by $6$ .

$\frac{30 + 30 i}{36 - 36 i + 36 i - 36 i i}$

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

$\frac{30 + 30 i}{36 - 36 i + 36 i - 36 (i^{1} i)}$

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

$\frac{30 + 30 i}{36 - 36 i + 36 i - 36 (i^{1} i^{1})}$

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

$\frac{30 + 30 i}{36 - 36 i + 36 i - 36 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{30 + 30 i}{36 - 36 i + 36 i - 36 i^{2}}$

Add $- 36 i$ and $36 i$ .

$\frac{30 + 30 i}{36 + 0 - 36 i^{2}}$

Add $36$ and $0$ .

$\frac{30 + 30 i}{36 - 36 i^{2}}$

Simplify each term.

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

$\frac{30 + 30 i}{36 - 36 \cdot - 1}$

Multiply $- 36$ by $- 1$ .

$\frac{30 + 30 i}{36 + 36}$

Add $36$ and $36$ .

$\frac{30 + 30 i}{72}$

Step 3

Cancel the common factor of $30 + 30 i$ and $72$ .

Factor $6$ out of $30$ .

$\frac{6 (5) + 30 i}{72}$

Factor $6$ out of $30 i$ .

$\frac{6 (5) + 6 (5 i)}{72}$

Factor $6$ out of $6 (5) + 6 (5 i)$ .

$\frac{6 (5 + 5 i)}{72}$

Cancel the common factors.

Factor $6$ out of $72$ .

$\frac{6 (5 + 5 i)}{6 \cdot 12}$

Cancel the common factor.

$\frac{6 (5 + 5 i)}{6 \cdot 12}$

Rewrite the expression.

$\frac{5 + 5 i}{12}$

Step 4

Split the fraction $\frac{5 + 5 i}{12}$ into two fractions.

$\frac{5}{12} + \frac{5 i}{12}$

Write in Standard Form (5i)/(6+6i)

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