Basic Math Examples

$\frac{6 + i}{2 + i}$

Step 1

Multiply the numerator and denominator of

$\frac{6 + i}{2 + 1 i}$ by the conjugate of

$2 + 1 i$ to make the denominator real.

$\frac{6 + i}{2 + 1 i} \cdot \frac{2 - i}{2 - i}$

Step 2

Multiply.

Tap for more steps...

Step 2.1

Combine.

$\frac{(6 + i) (2 - i)}{(2 + 1 i) (2 - i)}$

Step 2.2

Simplify the numerator.

Tap for more steps...

Step 2.2.1

Expand

$(6 + i) (2 - i)$ using the FOIL Method.

Tap for more steps...

Step 2.2.1.1

Apply the distributive property.

$\frac{6 (2 - i) + i (2 - i)}{(2 + 1 i) (2 - i)}$

Step 2.2.1.2

Apply the distributive property.

$\frac{6 \cdot 2 + 6 (- i) + i (2 - i)}{(2 + 1 i) (2 - i)}$

Step 2.2.1.3

Apply the distributive property.

$\frac{6 \cdot 2 + 6 (- i) + i \cdot 2 + i (- i)}{(2 + 1 i) (2 - i)}$

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

$2$ .

$\frac{12 + 6 (- i) + i \cdot 2 + i (- i)}{(2 + 1 i) (2 - i)}$

Step 2.2.2.1.2

Multiply

$- 1$ by

$6$ .

$\frac{12 - 6 i + i \cdot 2 + i (- i)}{(2 + 1 i) (2 - i)}$

Step 2.2.2.1.3

Move

$2$ to the left of

$i$ .

$\frac{12 - 6 i + 2 \cdot i + i (- i)}{(2 + 1 i) (2 - i)}$

Step 2.2.2.1.4

Multiply

$i (- i)$ .

Tap for more steps...

Step 2.2.2.1.4.1

Raise

$i$ to the power of

$1$ .

$\frac{12 - 6 i + 2 i - (i^{1} i)}{(2 + 1 i) (2 - i)}$

Step 2.2.2.1.4.2

Raise

$i$ to the power of

$1$ .

$\frac{12 - 6 i + 2 i - (i^{1} i^{1})}{(2 + 1 i) (2 - i)}$

Step 2.2.2.1.4.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{12 - 6 i + 2 i - i^{1 + 1}}{(2 + 1 i) (2 - i)}$

Step 2.2.2.1.4.4

Add

$1$ and

$1$ .

$\frac{12 - 6 i + 2 i - i^{2}}{(2 + 1 i) (2 - i)}$

Step 2.2.2.1.5

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{12 - 6 i + 2 i - - 1}{(2 + 1 i) (2 - i)}$

Step 2.2.2.1.6

Multiply

$- 1$ by

$- 1$ .

$\frac{12 - 6 i + 2 i + 1}{(2 + 1 i) (2 - i)}$

Step 2.2.2.2

Add

$12$ and

$1$ .

$\frac{13 - 6 i + 2 i}{(2 + 1 i) (2 - i)}$

Step 2.2.2.3

Add

$- 6 i$ and

$2 i$ .

$\frac{13 - 4 i}{(2 + 1 i) (2 - i)}$

Step 2.3

Simplify the denominator.

Tap for more steps...

Step 2.3.1

Expand

$(2 + 1 i) (2 - i)$ using the FOIL Method.

Tap for more steps...

Step 2.3.1.1

Apply the distributive property.

$\frac{13 - 4 i}{2 (2 - i) + 1 i (2 - i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{13 - 4 i}{2 \cdot 2 + 2 (- i) + 1 i (2 - i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{13 - 4 i}{2 \cdot 2 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)}$

Step 2.3.2

Simplify.

Tap for more steps...

Step 2.3.2.1

Multiply

$2$ by

$2$ .

$\frac{13 - 4 i}{4 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)}$

Step 2.3.2.2

Multiply

$- 1$ by

$2$ .

$\frac{13 - 4 i}{4 - 2 i + 1 i \cdot 2 + 1 i (- i)}$

Step 2.3.2.3

Multiply

$2$ by

$1$ .

$\frac{13 - 4 i}{4 - 2 i + 2 i + 1 i (- i)}$

Step 2.3.2.4

Multiply

$- 1$ by

$1$ .

$\frac{13 - 4 i}{4 - 2 i + 2 i - i i}$

Step 2.3.2.5

Raise

$i$ to the power of

$1$ .

$\frac{13 - 4 i}{4 - 2 i + 2 i - (i^{1} i)}$

Step 2.3.2.6

Raise

$i$ to the power of

$1$ .

$\frac{13 - 4 i}{4 - 2 i + 2 i - (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{13 - 4 i}{4 - 2 i + 2 i - i^{1 + 1}}$

Step 2.3.2.8

Add

$1$ and

$1$ .

$\frac{13 - 4 i}{4 - 2 i + 2 i - i^{2}}$

Step 2.3.2.9

Add

$- 2 i$ and

$2 i$ .

$\frac{13 - 4 i}{4 + 0 - i^{2}}$

Step 2.3.2.10

Add

$4$ and

$0$ .

$\frac{13 - 4 i}{4 - i^{2}}$

Step 2.3.3

Simplify each term.

Tap for more steps...

Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{13 - 4 i}{4 - - 1}$

Step 2.3.3.2

Multiply

$- 1$ by

$- 1$ .

$\frac{13 - 4 i}{4 + 1}$

Step 2.3.4

Add

$4$ and

$1$ .

$\frac{13 - 4 i}{5}$

Step 3

Split the fraction

$\frac{13 - 4 i}{5}$ into two fractions.

$\frac{13}{5} + \frac{- 4 i}{5}$

Step 4

Move the negative in front of the fraction.

$\frac{13}{5} - \frac{4 i}{5}$