Simplify (5-4i)/(8+6i)

$\frac{5 - 4 i}{8 + 6 i}$

Step 1

Multiply the numerator and denominator of $\frac{5 - 4 i}{8 + 6 i}$ by the conjugate of $8 + 6 i$ to make the denominator real.

$\frac{5 - 4 i}{8 + 6 i} \cdot \frac{8 - 6 i}{8 - 6 i}$

Step 2

Multiply.

Combine.

$\frac{(5 - 4 i) (8 - 6 i)}{(8 + 6 i) (8 - 6 i)}$

Simplify the numerator.

Expand $(5 - 4 i) (8 - 6 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{5 (8 - 6 i) - 4 i (8 - 6 i)}{(8 + 6 i) (8 - 6 i)}$

Apply the distributive property.

$\frac{5 \cdot 8 + 5 (- 6 i) - 4 i (8 - 6 i)}{(8 + 6 i) (8 - 6 i)}$

Apply the distributive property.

$\frac{5 \cdot 8 + 5 (- 6 i) - 4 i \cdot 8 - 4 i (- 6 i)}{(8 + 6 i) (8 - 6 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $8$ .

$\frac{40 + 5 (- 6 i) - 4 i \cdot 8 - 4 i (- 6 i)}{(8 + 6 i) (8 - 6 i)}$

Multiply $- 6$ by $5$ .

$\frac{40 - 30 i - 4 i \cdot 8 - 4 i (- 6 i)}{(8 + 6 i) (8 - 6 i)}$

Multiply $8$ by $- 4$ .

$\frac{40 - 30 i - 32 i - 4 i (- 6 i)}{(8 + 6 i) (8 - 6 i)}$

Multiply $- 4 i (- 6 i)$ .

Multiply $- 6$ by $- 4$ .

$\frac{40 - 30 i - 32 i + 24 i i}{(8 + 6 i) (8 - 6 i)}$

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

$\frac{40 - 30 i - 32 i + 24 (i^{1} i)}{(8 + 6 i) (8 - 6 i)}$

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

$\frac{40 - 30 i - 32 i + 24 (i^{1} i^{1})}{(8 + 6 i) (8 - 6 i)}$

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

$\frac{40 - 30 i - 32 i + 24 i^{1 + 1}}{(8 + 6 i) (8 - 6 i)}$

Add $1$ and $1$ .

$\frac{40 - 30 i - 32 i + 24 i^{2}}{(8 + 6 i) (8 - 6 i)}$

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

$\frac{40 - 30 i - 32 i + 24 \cdot - 1}{(8 + 6 i) (8 - 6 i)}$

Multiply $24$ by $- 1$ .

$\frac{40 - 30 i - 32 i - 24}{(8 + 6 i) (8 - 6 i)}$

Subtract $24$ from $40$ .

$\frac{16 - 30 i - 32 i}{(8 + 6 i) (8 - 6 i)}$

Subtract $32 i$ from $- 30 i$ .

$\frac{16 - 62 i}{(8 + 6 i) (8 - 6 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{16 - 62 i}{8 (8 - 6 i) + 6 i (8 - 6 i)}$

Apply the distributive property.

$\frac{16 - 62 i}{8 \cdot 8 + 8 (- 6 i) + 6 i (8 - 6 i)}$

Apply the distributive property.

$\frac{16 - 62 i}{8 \cdot 8 + 8 (- 6 i) + 6 i \cdot 8 + 6 i (- 6 i)}$

Simplify.

Multiply $8$ by $8$ .

$\frac{16 - 62 i}{64 + 8 (- 6 i) + 6 i \cdot 8 + 6 i (- 6 i)}$

Multiply $- 6$ by $8$ .

$\frac{16 - 62 i}{64 - 48 i + 6 i \cdot 8 + 6 i (- 6 i)}$

Multiply $8$ by $6$ .

$\frac{16 - 62 i}{64 - 48 i + 48 i + 6 i (- 6 i)}$

Multiply $- 6$ by $6$ .

$\frac{16 - 62 i}{64 - 48 i + 48 i - 36 i i}$

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

$\frac{16 - 62 i}{64 - 48 i + 48 i - 36 (i^{1} i)}$

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

$\frac{16 - 62 i}{64 - 48 i + 48 i - 36 (i^{1} i^{1})}$

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

$\frac{16 - 62 i}{64 - 48 i + 48 i - 36 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{16 - 62 i}{64 - 48 i + 48 i - 36 i^{2}}$

Add $- 48 i$ and $48 i$ .

$\frac{16 - 62 i}{64 + 0 - 36 i^{2}}$

Add $64$ and $0$ .

$\frac{16 - 62 i}{64 - 36 i^{2}}$

Simplify each term.

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

$\frac{16 - 62 i}{64 - 36 \cdot - 1}$

Multiply $- 36$ by $- 1$ .

$\frac{16 - 62 i}{64 + 36}$

Add $64$ and $36$ .

$\frac{16 - 62 i}{100}$

Step 3

Cancel the common factor of $16 - 62 i$ and $100$ .

Factor $2$ out of $16$ .

$\frac{2 (8) - 62 i}{100}$

Factor $2$ out of $- 62 i$ .

$\frac{2 (8) + 2 (- 31 i)}{100}$

Factor $2$ out of $2 (8) + 2 (- 31 i)$ .

$\frac{2 (8 - 31 i)}{100}$

Cancel the common factors.

Factor $2$ out of $100$ .

$\frac{2 (8 - 31 i)}{2 \cdot 50}$

Cancel the common factor.

$\frac{2 (8 - 31 i)}{2 \cdot 50}$

Rewrite the expression.

$\frac{8 - 31 i}{50}$

Step 4

Split the fraction $\frac{8 - 31 i}{50}$ into two fractions.

$\frac{8}{50} + \frac{- 31 i}{50}$

Step 5

Simplify each term.

Cancel the common factor of $8$ and $50$ .

Factor $2$ out of $8$ .

$\frac{2 (4)}{50} + \frac{- 31 i}{50}$

Cancel the common factors.

Factor $2$ out of $50$ .

$\frac{2 \cdot 4}{2 \cdot 25} + \frac{- 31 i}{50}$

Cancel the common factor.

$\frac{2 \cdot 4}{2 \cdot 25} + \frac{- 31 i}{50}$

Rewrite the expression.

$\frac{4}{25} + \frac{- 31 i}{50}$

Move the negative in front of the fraction.

$\frac{4}{25} - \frac{31 i}{50}$

Simplify (5-4i)/(8+6i)

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