Evaluate (8+i)/(8-i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $8$ by $8$ .

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

Move $8$ to the left of $i$ .

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

Multiply $i i$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Subtract $1$ from $64$ .

$\frac{63 + 8 i + 8 i}{(8 - i) (8 + i)}$

Add $8 i$ and $8 i$ .

$\frac{63 + 16 i}{(8 - i) (8 + i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{63 + 16 i}{8 (8 + i) - i (8 + i)}$

Apply the distributive property.

$\frac{63 + 16 i}{8 \cdot 8 + 8 i - i (8 + i)}$

Apply the distributive property.

$\frac{63 + 16 i}{8 \cdot 8 + 8 i - i \cdot 8 - i i}$

Simplify.

Multiply $8$ by $8$ .

$\frac{63 + 16 i}{64 + 8 i - i \cdot 8 - i i}$

Multiply $8$ by $- 1$ .

$\frac{63 + 16 i}{64 + 8 i - 8 i - i i}$

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

$\frac{63 + 16 i}{64 + 8 i - 8 i - (i^{1} i)}$

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

$\frac{63 + 16 i}{64 + 8 i - 8 i - (i^{1} i^{1})}$

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

$\frac{63 + 16 i}{64 + 8 i - 8 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{63 + 16 i}{64 + 8 i - 8 i - i^{2}}$

Subtract $8 i$ from $8 i$ .

$\frac{63 + 16 i}{64 + 0 - i^{2}}$

Add $64$ and $0$ .

$\frac{63 + 16 i}{64 - i^{2}}$

Simplify each term.

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

$\frac{63 + 16 i}{64 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{63 + 16 i}{64 + 1}$

Add $64$ and $1$ .

$\frac{63 + 16 i}{65}$

Step 3

Split the fraction $\frac{63 + 16 i}{65}$ into two fractions.

$\frac{63}{65} + \frac{16 i}{65}$

Evaluate (8+i)/(8-i)

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