Evaluate (4+i)/(4-i)

$\frac{4 + i}{4 - i}$

Step 1

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

$\frac{4 + i}{4 - i} \cdot \frac{4 + i}{4 + i}$

Step 2

Multiply.

Combine.

$\frac{(4 + i) (4 + i)}{(4 - i) (4 + i)}$

Simplify the numerator.

Expand $(4 + i) (4 + i)$ using the FOIL Method.

Apply the distributive property.

$\frac{4 (4 + i) + i (4 + i)}{(4 - i) (4 + i)}$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$\frac{16 + 4 i + i \cdot 4 + i i}{(4 - i) (4 + i)}$

Move $4$ to the left of $i$ .

$\frac{16 + 4 i + 4 \cdot i + i i}{(4 - i) (4 + i)}$

Multiply $i i$ .

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

$\frac{16 + 4 i + 4 i + i^{1} i}{(4 - i) (4 + i)}$

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

$\frac{16 + 4 i + 4 i + i^{1} i^{1}}{(4 - i) (4 + i)}$

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

$\frac{16 + 4 i + 4 i + i^{1 + 1}}{(4 - i) (4 + i)}$

Add $1$ and $1$ .

$\frac{16 + 4 i + 4 i + i^{2}}{(4 - i) (4 + i)}$

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

$\frac{16 + 4 i + 4 i - 1}{(4 - i) (4 + i)}$

Subtract $1$ from $16$ .

$\frac{15 + 4 i + 4 i}{(4 - i) (4 + i)}$

Add $4 i$ and $4 i$ .

$\frac{15 + 8 i}{(4 - i) (4 + i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{15 + 8 i}{4 (4 + i) - i (4 + i)}$

Apply the distributive property.

$\frac{15 + 8 i}{4 \cdot 4 + 4 i - i (4 + i)}$

Apply the distributive property.

$\frac{15 + 8 i}{4 \cdot 4 + 4 i - i \cdot 4 - i i}$

Simplify.

Multiply $4$ by $4$ .

$\frac{15 + 8 i}{16 + 4 i - i \cdot 4 - i i}$

Multiply $4$ by $- 1$ .

$\frac{15 + 8 i}{16 + 4 i - 4 i - i i}$

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

$\frac{15 + 8 i}{16 + 4 i - 4 i - (i^{1} i)}$

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

$\frac{15 + 8 i}{16 + 4 i - 4 i - (i^{1} i^{1})}$

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

$\frac{15 + 8 i}{16 + 4 i - 4 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{15 + 8 i}{16 + 4 i - 4 i - i^{2}}$

Subtract $4 i$ from $4 i$ .

$\frac{15 + 8 i}{16 + 0 - i^{2}}$

Add $16$ and $0$ .

$\frac{15 + 8 i}{16 - i^{2}}$

Simplify each term.

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

$\frac{15 + 8 i}{16 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{15 + 8 i}{16 + 1}$

Add $16$ and $1$ .

$\frac{15 + 8 i}{17}$

Step 3

Split the fraction $\frac{15 + 8 i}{17}$ into two fractions.

$\frac{15}{17} + \frac{8 i}{17}$

Evaluate (4+i)/(4-i)

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