Write in Standard Form (6+3i)/(8-6i)

$\frac{6 + 3 i}{8 - 6 i}$

Step 1

Multiply the numerator and denominator of $\frac{6 + 3 i}{8 - 6 i}$ by the conjugate of $8 - 6 i$ to make the denominator real.

$\frac{6 + 3 i}{8 - 6 i} \cdot \frac{8 + 6 i}{8 + 6 i}$

Step 2

Multiply.

Combine.

$\frac{(6 + 3 i) (8 + 6 i)}{(8 - 6 i) (8 + 6 i)}$

Simplify the numerator.

Expand $(6 + 3 i) (8 + 6 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{6 (8 + 6 i) + 3 i (8 + 6 i)}{(8 - 6 i) (8 + 6 i)}$

Apply the distributive property.

$\frac{6 \cdot 8 + 6 (6 i) + 3 i (8 + 6 i)}{(8 - 6 i) (8 + 6 i)}$

Apply the distributive property.

$\frac{6 \cdot 8 + 6 (6 i) + 3 i \cdot 8 + 3 i (6 i)}{(8 - 6 i) (8 + 6 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $8$ .

$\frac{48 + 6 (6 i) + 3 i \cdot 8 + 3 i (6 i)}{(8 - 6 i) (8 + 6 i)}$

Multiply $6$ by $6$ .

$\frac{48 + 36 i + 3 i \cdot 8 + 3 i (6 i)}{(8 - 6 i) (8 + 6 i)}$

Multiply $8$ by $3$ .

$\frac{48 + 36 i + 24 i + 3 i (6 i)}{(8 - 6 i) (8 + 6 i)}$

Multiply $3 i (6 i)$ .

Multiply $6$ by $3$ .

$\frac{48 + 36 i + 24 i + 18 i i}{(8 - 6 i) (8 + 6 i)}$

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

$\frac{48 + 36 i + 24 i + 18 (i^{1} i)}{(8 - 6 i) (8 + 6 i)}$

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

$\frac{48 + 36 i + 24 i + 18 (i^{1} i^{1})}{(8 - 6 i) (8 + 6 i)}$

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

$\frac{48 + 36 i + 24 i + 18 i^{1 + 1}}{(8 - 6 i) (8 + 6 i)}$

Add $1$ and $1$ .

$\frac{48 + 36 i + 24 i + 18 i^{2}}{(8 - 6 i) (8 + 6 i)}$

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

$\frac{48 + 36 i + 24 i + 18 \cdot - 1}{(8 - 6 i) (8 + 6 i)}$

Multiply $18$ by $- 1$ .

$\frac{48 + 36 i + 24 i - 18}{(8 - 6 i) (8 + 6 i)}$

Subtract $18$ from $48$ .

$\frac{30 + 36 i + 24 i}{(8 - 6 i) (8 + 6 i)}$

Add $36 i$ and $24 i$ .

$\frac{30 + 60 i}{(8 - 6 i) (8 + 6 i)}$

Simplify the denominator.

Expand $(8 - 6 i) (8 + 6 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{30 + 60 i}{8 (8 + 6 i) - 6 i (8 + 6 i)}$

Apply the distributive property.

$\frac{30 + 60 i}{8 \cdot 8 + 8 (6 i) - 6 i (8 + 6 i)}$

Apply the distributive property.

$\frac{30 + 60 i}{8 \cdot 8 + 8 (6 i) - 6 i \cdot 8 - 6 i (6 i)}$

Simplify.

Multiply $8$ by $8$ .

$\frac{30 + 60 i}{64 + 8 (6 i) - 6 i \cdot 8 - 6 i (6 i)}$

Multiply $6$ by $8$ .

$\frac{30 + 60 i}{64 + 48 i - 6 i \cdot 8 - 6 i (6 i)}$

Multiply $8$ by $- 6$ .

$\frac{30 + 60 i}{64 + 48 i - 48 i - 6 i (6 i)}$

Multiply $6$ by $- 6$ .

$\frac{30 + 60 i}{64 + 48 i - 48 i - 36 i i}$

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

$\frac{30 + 60 i}{64 + 48 i - 48 i - 36 (i^{1} i)}$

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

$\frac{30 + 60 i}{64 + 48 i - 48 i - 36 (i^{1} i^{1})}$

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

$\frac{30 + 60 i}{64 + 48 i - 48 i - 36 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{30 + 60 i}{64 + 48 i - 48 i - 36 i^{2}}$

Subtract $48 i$ from $48 i$ .

$\frac{30 + 60 i}{64 + 0 - 36 i^{2}}$

Add $64$ and $0$ .

$\frac{30 + 60 i}{64 - 36 i^{2}}$

Simplify each term.

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

$\frac{30 + 60 i}{64 - 36 \cdot - 1}$

Multiply $- 36$ by $- 1$ .

$\frac{30 + 60 i}{64 + 36}$

Add $64$ and $36$ .

$\frac{30 + 60 i}{100}$

Step 3

Cancel the common factor of $30 + 60 i$ and $100$ .

Factor $10$ out of $30$ .

$\frac{10 (3) + 60 i}{100}$

Factor $10$ out of $60 i$ .

$\frac{10 (3) + 10 (6 i)}{100}$

Factor $10$ out of $10 (3) + 10 (6 i)$ .

$\frac{10 (3 + 6 i)}{100}$

Cancel the common factors.

Factor $10$ out of $100$ .

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

Cancel the common factor.

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

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

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

Factor $2$ out of $6 i$ .

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

Cancel the common factors.

Factor $2$ out of $10$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Write in Standard Form (6+3i)/(8-6i)

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