Simplify (8-7i)/(8+i)

$\frac{8 - 7 i}{8 + i}$

Multiply the numerator and denominator of $\frac{8 - 7 i}{8 + 1 i}$ by the conjugate of $8 + 1 i$ to make the denominator real.

$\frac{8 - 7 i}{8 + 1 i} \cdot \frac{8 - i}{8 - i}$

Multiply.

Combine.

$\frac{(8 - 7 i) (8 - i)}{(8 + 1 i) (8 - i)}$

Simplify the numerator.

Expand $(8 - 7 i) (8 - i)$ using the FOIL Method.

Apply the distributive property.

$\frac{8 (8 - i) - 7 i (8 - i)}{(8 + 1 i) (8 - i)}$

Apply the distributive property.

$\frac{8 \cdot 8 + 8 (- i) - 7 i (8 - i)}{(8 + 1 i) (8 - i)}$

Apply the distributive property.

$\frac{8 \cdot 8 + 8 (- i) - 7 i \cdot 8 - 7 i (- i)}{(8 + 1 i) (8 - i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $8$ by $8$ .

$\frac{64 + 8 (- i) - 7 i \cdot 8 - 7 i (- i)}{(8 + 1 i) (8 - i)}$

Multiply $- 1$ by $8$ .

$\frac{64 - 8 i - 7 i \cdot 8 - 7 i (- i)}{(8 + 1 i) (8 - i)}$

Multiply $8$ by $- 7$ .

$\frac{64 - 8 i - 56 i - 7 i (- i)}{(8 + 1 i) (8 - i)}$

Multiply $- 7 i (- i)$ .

Multiply $- 1$ by $- 7$ .

$\frac{64 - 8 i - 56 i + 7 i i}{(8 + 1 i) (8 - i)}$

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

$\frac{64 - 8 i - 56 i + 7 (i^{1} i)}{(8 + 1 i) (8 - i)}$

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

$\frac{64 - 8 i - 56 i + 7 (i^{1} i^{1})}{(8 + 1 i) (8 - i)}$

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

$\frac{64 - 8 i - 56 i + 7 i^{1 + 1}}{(8 + 1 i) (8 - i)}$

Add $1$ and $1$ .

$\frac{64 - 8 i - 56 i + 7 i^{2}}{(8 + 1 i) (8 - i)}$

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

$\frac{64 - 8 i - 56 i + 7 \cdot - 1}{(8 + 1 i) (8 - i)}$

Multiply $7$ by $- 1$ .

$\frac{64 - 8 i - 56 i - 7}{(8 + 1 i) (8 - i)}$

Subtract $7$ from $64$ .

$\frac{57 - 8 i - 56 i}{(8 + 1 i) (8 - i)}$

Subtract $56 i$ from $- 8 i$ .

$\frac{57 - 64 i}{(8 + 1 i) (8 - i)}$

Simplify the denominator.

Expand $(8 + 1 i) (8 - i)$ using the FOIL Method.

Apply the distributive property.

$\frac{57 - 64 i}{8 (8 - i) + 1 i (8 - i)}$

Apply the distributive property.

$\frac{57 - 64 i}{8 \cdot 8 + 8 (- i) + 1 i (8 - i)}$

Apply the distributive property.

$\frac{57 - 64 i}{8 \cdot 8 + 8 (- i) + 1 i \cdot 8 + 1 i (- i)}$

Simplify.

Multiply $8$ by $8$ .

$\frac{57 - 64 i}{64 + 8 (- i) + 1 i \cdot 8 + 1 i (- i)}$

Multiply $- 1$ by $8$ .

$\frac{57 - 64 i}{64 - 8 i + 1 i \cdot 8 + 1 i (- i)}$

Multiply $8$ by $1$ .

$\frac{57 - 64 i}{64 - 8 i + 8 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{57 - 64 i}{64 - 8 i + 8 i - i i}$

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

$\frac{57 - 64 i}{64 - 8 i + 8 i - (i^{1} i)}$

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

$\frac{57 - 64 i}{64 - 8 i + 8 i - (i^{1} i^{1})}$

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

$\frac{57 - 64 i}{64 - 8 i + 8 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{57 - 64 i}{64 - 8 i + 8 i - i^{2}}$

Add $- 8 i$ and $8 i$ .

$\frac{57 - 64 i}{64 + 0 - i^{2}}$

Add $64$ and $0$ .

$\frac{57 - 64 i}{64 - i^{2}}$

Simplify each term.

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

$\frac{57 - 64 i}{64 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{57 - 64 i}{64 + 1}$

Add $64$ and $1$ .

$\frac{57 - 64 i}{65}$

Split the fraction $\frac{57 - 64 i}{65}$ into two fractions.

$\frac{57}{65} + \frac{- 64 i}{65}$

Move the negative in front of the fraction.

$\frac{57}{65} - \frac{64 i}{65}$

Simplify (8-7i)/(8+i)

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