Precalculus Examples

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

Step 1

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

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

Step 2

Multiply.

Tap for more steps...

Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

Tap for more steps...

Step 2.2.1

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

Tap for more steps...

Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

Step 2.2.2

Simplify and combine like terms.

Tap for more steps...

Step 2.2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.2.1.1

Multiply $6$ by $4$ .

$\frac{24 + 6 i + i \cdot 4 + i i}{(4 - i) (4 + i)}$

Step 2.2.2.1.2

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

$\frac{24 + 6 i + 4 \cdot i + i i}{(4 - i) (4 + i)}$

Step 2.2.2.1.3

Multiply $i i$ .

Tap for more steps...

Step 2.2.2.1.3.1

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

$\frac{24 + 6 i + 4 i + i^{1} i}{(4 - i) (4 + i)}$

Step 2.2.2.1.3.2

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

$\frac{24 + 6 i + 4 i + i^{1} i^{1}}{(4 - i) (4 + i)}$

Step 2.2.2.1.3.3

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

$\frac{24 + 6 i + 4 i + i^{1 + 1}}{(4 - i) (4 + i)}$

Step 2.2.2.1.3.4

Add $1$ and $1$ .

$\frac{24 + 6 i + 4 i + i^{2}}{(4 - i) (4 + i)}$

Step 2.2.2.1.4

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

$\frac{24 + 6 i + 4 i - 1}{(4 - i) (4 + i)}$

Step 2.2.2.2

Subtract $1$ from $24$ .

$\frac{23 + 6 i + 4 i}{(4 - i) (4 + i)}$

Step 2.2.2.3

Add $6 i$ and $4 i$ .

$\frac{23 + 10 i}{(4 - i) (4 + i)}$

Step 2.3

Simplify the denominator.

Tap for more steps...

Step 2.3.1

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

Tap for more steps...

Step 2.3.1.1

Apply the distributive property.

$\frac{23 + 10 i}{4 (4 + i) - i (4 + i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{23 + 10 i}{4 \cdot 4 + 4 i - i (4 + i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{23 + 10 i}{4 \cdot 4 + 4 i - i \cdot 4 - i i}$

Step 2.3.2

Simplify.

Tap for more steps...

Step 2.3.2.1

Multiply $4$ by $4$ .

$\frac{23 + 10 i}{16 + 4 i - i \cdot 4 - i i}$

Step 2.3.2.2

Multiply $4$ by $- 1$ .

$\frac{23 + 10 i}{16 + 4 i - 4 i - i i}$

Step 2.3.2.3

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

$\frac{23 + 10 i}{16 + 4 i - 4 i - (i^{1} i)}$

Step 2.3.2.4

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

$\frac{23 + 10 i}{16 + 4 i - 4 i - (i^{1} i^{1})}$

Step 2.3.2.5

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

$\frac{23 + 10 i}{16 + 4 i - 4 i - i^{1 + 1}}$

Step 2.3.2.6

Add $1$ and $1$ .

$\frac{23 + 10 i}{16 + 4 i - 4 i - i^{2}}$

Step 2.3.2.7

Subtract $4 i$ from $4 i$ .

$\frac{23 + 10 i}{16 + 0 - i^{2}}$

Step 2.3.2.8

Add $16$ and $0$ .

$\frac{23 + 10 i}{16 - i^{2}}$

Step 2.3.3

Simplify each term.

Tap for more steps...

Step 2.3.3.1

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

$\frac{23 + 10 i}{16 - - 1}$

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{23 + 10 i}{16 + 1}$

Step 2.3.4

Add $16$ and $1$ .

$\frac{23 + 10 i}{17}$

Step 3

Split the fraction $\frac{23 + 10 i}{17}$ into two fractions.

$\frac{23}{17} + \frac{10 i}{17}$