Evaluate (5+4i)/(6+8i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $6$ .

$\frac{30 + 5 (- 8 i) + 4 i \cdot 6 + 4 i (- 8 i)}{(6 + 8 i) (6 - 8 i)}$

Multiply $- 8$ by $5$ .

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

Multiply $6$ by $4$ .

$\frac{30 - 40 i + 24 i + 4 i (- 8 i)}{(6 + 8 i) (6 - 8 i)}$

Multiply $4 i (- 8 i)$ .

Multiply $- 8$ by $4$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 32$ by $- 1$ .

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

Add $30$ and $32$ .

$\frac{62 - 40 i + 24 i}{(6 + 8 i) (6 - 8 i)}$

Add $- 40 i$ and $24 i$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $6$ by $6$ .

$\frac{62 - 16 i}{36 + 6 (- 8 i) + 8 i \cdot 6 + 8 i (- 8 i)}$

Multiply $- 8$ by $6$ .

$\frac{62 - 16 i}{36 - 48 i + 8 i \cdot 6 + 8 i (- 8 i)}$

Multiply $6$ by $8$ .

$\frac{62 - 16 i}{36 - 48 i + 48 i + 8 i (- 8 i)}$

Multiply $- 8$ by $8$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Add $36$ and $0$ .

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

Simplify each term.

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

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

Multiply $- 64$ by $- 1$ .

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

Add $36$ and $64$ .

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

Step 3

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

Factor $2$ out of $62$ .

$\frac{2 (31) - 16 i}{100}$

Factor $2$ out of $- 16 i$ .

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

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

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

Cancel the common factors.

Factor $2$ out of $100$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

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

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

Step 5

Simplify each term.

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

Factor $2$ out of $- 8 i$ .

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

Cancel the common factors.

Factor $2$ out of $50$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Evaluate (5+4i)/(6+8i)

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