Basic Math Examples

$\frac{3 - i}{2 + 3 i}$

Step 1

Multiply the numerator and denominator of

$\frac{3 - i}{2 + 3 i}$ by the conjugate of

$2 + 3 i$ to make the denominator real.

$\frac{3 - i}{2 + 3 i} \cdot \frac{2 - 3 i}{2 - 3 i}$

Step 2

Multiply.

Tap for more steps...

Step 2.1

Combine.

$\frac{(3 - i) (2 - 3 i)}{(2 + 3 i) (2 - 3 i)}$

Step 2.2

Simplify the numerator.

Tap for more steps...

Step 2.2.1

Expand

$(3 - i) (2 - 3 i)$ using the FOIL Method.

Tap for more steps...

Step 2.2.1.1

Apply the distributive property.

$\frac{3 (2 - 3 i) - i (2 - 3 i)}{(2 + 3 i) (2 - 3 i)}$

Step 2.2.1.2

Apply the distributive property.

$\frac{3 \cdot 2 + 3 (- 3 i) - i (2 - 3 i)}{(2 + 3 i) (2 - 3 i)}$

Step 2.2.1.3

Apply the distributive property.

$\frac{3 \cdot 2 + 3 (- 3 i) - i \cdot 2 - i (- 3 i)}{(2 + 3 i) (2 - 3 i)}$

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

$3$ by

$2$ .

$\frac{6 + 3 (- 3 i) - i \cdot 2 - i (- 3 i)}{(2 + 3 i) (2 - 3 i)}$

Step 2.2.2.1.2

Multiply

$- 3$ by

$3$ .

$\frac{6 - 9 i - i \cdot 2 - i (- 3 i)}{(2 + 3 i) (2 - 3 i)}$

Step 2.2.2.1.3

Multiply

$2$ by

$- 1$ .

$\frac{6 - 9 i - 2 i - i (- 3 i)}{(2 + 3 i) (2 - 3 i)}$

Step 2.2.2.1.4

Multiply

$- i (- 3 i)$ .

Tap for more steps...

Step 2.2.2.1.4.1

Multiply

$- 3$ by

$- 1$ .

$\frac{6 - 9 i - 2 i + 3 i i}{(2 + 3 i) (2 - 3 i)}$

Step 2.2.2.1.4.2

Raise

$i$ to the power of

$1$ .

$\frac{6 - 9 i - 2 i + 3 (i^{1} i)}{(2 + 3 i) (2 - 3 i)}$

Step 2.2.2.1.4.3

Raise

$i$ to the power of

$1$ .

$\frac{6 - 9 i - 2 i + 3 (i^{1} i^{1})}{(2 + 3 i) (2 - 3 i)}$

Step 2.2.2.1.4.4

Use the power rule

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

$\frac{6 - 9 i - 2 i + 3 i^{1 + 1}}{(2 + 3 i) (2 - 3 i)}$

Step 2.2.2.1.4.5

Add

$1$ and

$1$ .

$\frac{6 - 9 i - 2 i + 3 i^{2}}{(2 + 3 i) (2 - 3 i)}$

Step 2.2.2.1.5

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{6 - 9 i - 2 i + 3 \cdot - 1}{(2 + 3 i) (2 - 3 i)}$

Step 2.2.2.1.6

Multiply

$3$ by

$- 1$ .

$\frac{6 - 9 i - 2 i - 3}{(2 + 3 i) (2 - 3 i)}$

Step 2.2.2.2

Subtract

$3$ from

$6$ .

$\frac{3 - 9 i - 2 i}{(2 + 3 i) (2 - 3 i)}$

Step 2.2.2.3

Subtract

$2 i$ from

$- 9 i$ .

$\frac{3 - 11 i}{(2 + 3 i) (2 - 3 i)}$

Step 2.3

Simplify the denominator.

Tap for more steps...

Step 2.3.1

Expand

$(2 + 3 i) (2 - 3 i)$ using the FOIL Method.

Tap for more steps...

Step 2.3.1.1

Apply the distributive property.

$\frac{3 - 11 i}{2 (2 - 3 i) + 3 i (2 - 3 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{3 - 11 i}{2 \cdot 2 + 2 (- 3 i) + 3 i (2 - 3 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{3 - 11 i}{2 \cdot 2 + 2 (- 3 i) + 3 i \cdot 2 + 3 i (- 3 i)}$

Step 2.3.2

Simplify.

Tap for more steps...

Step 2.3.2.1

Multiply

$2$ by

$2$ .

$\frac{3 - 11 i}{4 + 2 (- 3 i) + 3 i \cdot 2 + 3 i (- 3 i)}$

Step 2.3.2.2

Multiply

$- 3$ by

$2$ .

$\frac{3 - 11 i}{4 - 6 i + 3 i \cdot 2 + 3 i (- 3 i)}$

Step 2.3.2.3

Multiply

$2$ by

$3$ .

$\frac{3 - 11 i}{4 - 6 i + 6 i + 3 i (- 3 i)}$

Step 2.3.2.4

Multiply

$- 3$ by

$3$ .

$\frac{3 - 11 i}{4 - 6 i + 6 i - 9 i i}$

Step 2.3.2.5

Raise

$i$ to the power of

$1$ .

$\frac{3 - 11 i}{4 - 6 i + 6 i - 9 (i^{1} i)}$

Step 2.3.2.6

Raise

$i$ to the power of

$1$ .

$\frac{3 - 11 i}{4 - 6 i + 6 i - 9 (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

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

$\frac{3 - 11 i}{4 - 6 i + 6 i - 9 i^{1 + 1}}$

Step 2.3.2.8

Add

$1$ and

$1$ .

$\frac{3 - 11 i}{4 - 6 i + 6 i - 9 i^{2}}$

Step 2.3.2.9

Add

$- 6 i$ and

$6 i$ .

$\frac{3 - 11 i}{4 + 0 - 9 i^{2}}$

Step 2.3.2.10

Add

$4$ and

$0$ .

$\frac{3 - 11 i}{4 - 9 i^{2}}$

Step 2.3.3

Simplify each term.

Tap for more steps...

Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{3 - 11 i}{4 - 9 \cdot - 1}$

Step 2.3.3.2

Multiply

$- 9$ by

$- 1$ .

$\frac{3 - 11 i}{4 + 9}$

Step 2.3.4

Add

$4$ and

$9$ .

$\frac{3 - 11 i}{13}$

Step 3

Split the fraction

$\frac{3 - 11 i}{13}$ into two fractions.

$\frac{3}{13} + \frac{- 11 i}{13}$

Step 4

Move the negative in front of the fraction.

$\frac{3}{13} - \frac{11 i}{13}$