Basic Math Examples

\frac{4}{9 - i}

$\frac{4}{9 - i}$

Step 1

Multiply the numerator and denominator of

\frac{4}{9 - i}

$\frac{4}{9 - i}$ by the conjugate of

9 - i

$9 - i$ to make the denominator real.

\frac{4}{9 - i} \cdot \frac{9 + i}{9 + i}

$\frac{4}{9 - i} \cdot \frac{9 + i}{9 + i}$

Step 2

Multiply.

Tap for more steps...

Step 2.1

Combine.

\frac{4 (9 + i)}{(9 - i) (9 + i)}

$\frac{4 (9 + i)}{(9 - i) (9 + i)}$

Step 2.2

Simplify the numerator.

Tap for more steps...

Step 2.2.1

Apply the distributive property.

\frac{4 \cdot 9 + 4 i}{(9 - i) (9 + i)}

$\frac{4 \cdot 9 + 4 i}{(9 - i) (9 + i)}$

Step 2.2.2

Multiply

4

$4$ by

9

$9$ .

\frac{36 + 4 i}{(9 - i) (9 + i)}

$\frac{36 + 4 i}{(9 - i) (9 + i)}$

\frac{36 + 4 i}{(9 - i) (9 + i)}

$\frac{36 + 4 i}{(9 - i) (9 + i)}$

Step 2.3

Simplify the denominator.

Tap for more steps...

Step 2.3.1

Expand

(9 - i) (9 + i)

$(9 - i) (9 + i)$ using the FOIL Method.

Tap for more steps...

Step 2.3.1.1

Apply the distributive property.

\frac{36 + 4 i}{9 (9 + i) - i (9 + i)}

$\frac{36 + 4 i}{9 (9 + i) - i (9 + i)}$

Step 2.3.1.2

Apply the distributive property.

\frac{36 + 4 i}{9 \cdot 9 + 9 i - i (9 + i)}

$\frac{36 + 4 i}{9 \cdot 9 + 9 i - i (9 + i)}$

Step 2.3.1.3

Apply the distributive property.

\frac{36 + 4 i}{9 \cdot 9 + 9 i - i \cdot 9 - i i}

$\frac{36 + 4 i}{9 \cdot 9 + 9 i - i \cdot 9 - i i}$

\frac{36 + 4 i}{9 \cdot 9 + 9 i - i \cdot 9 - i i}

$\frac{36 + 4 i}{9 \cdot 9 + 9 i - i \cdot 9 - i i}$

Step 2.3.2

Simplify.

Tap for more steps...

Step 2.3.2.1

Multiply

9

$9$ by

9

$9$ .

\frac{36 + 4 i}{81 + 9 i - i \cdot 9 - i i}

$\frac{36 + 4 i}{81 + 9 i - i \cdot 9 - i i}$

Step 2.3.2.2

Multiply

9

$9$ by

- 1

$- 1$ .

\frac{36 + 4 i}{81 + 9 i - 9 i - i i}

$\frac{36 + 4 i}{81 + 9 i - 9 i - i i}$

Step 2.3.2.3

Raise

i

$i$ to the power of

1

$1$ .

\frac{36 + 4 i}{81 + 9 i - 9 i - (i^{1} i)}

$\frac{36 + 4 i}{81 + 9 i - 9 i - (i^{1} i)}$

Step 2.3.2.4

Raise

i

$i$ to the power of

1

$1$ .

\frac{36 + 4 i}{81 + 9 i - 9 i - (i^{1} i^{1})}

$\frac{36 + 4 i}{81 + 9 i - 9 i - (i^{1} i^{1})}$

Step 2.3.2.5

Use the power rule

a^{m} a^{n} = a^{m + n}

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

\frac{36 + 4 i}{81 + 9 i - 9 i - i^{1 + 1}}

$\frac{36 + 4 i}{81 + 9 i - 9 i - i^{1 + 1}}$

Step 2.3.2.6

Add

1

$1$ and

1

$1$ .

\frac{36 + 4 i}{81 + 9 i - 9 i - i^{2}}

$\frac{36 + 4 i}{81 + 9 i - 9 i - i^{2}}$

Step 2.3.2.7

Subtract

9 i

$9 i$ from

9 i

$9 i$ .

\frac{36 + 4 i}{81 + 0 - i^{2}}

$\frac{36 + 4 i}{81 + 0 - i^{2}}$

Step 2.3.2.8

Add

81

$81$ and

0

$0$ .

\frac{36 + 4 i}{81 - i^{2}}

$\frac{36 + 4 i}{81 - i^{2}}$

\frac{36 + 4 i}{81 - i^{2}}

$\frac{36 + 4 i}{81 - i^{2}}$

Step 2.3.3

Simplify each term.

Tap for more steps...

Step 2.3.3.1

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{36 + 4 i}{81 - - 1}

$\frac{36 + 4 i}{81 - - 1}$

Step 2.3.3.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{36 + 4 i}{81 + 1}

$\frac{36 + 4 i}{81 + 1}$

\frac{36 + 4 i}{81 + 1}

$\frac{36 + 4 i}{81 + 1}$

Step 2.3.4

Add

81

$81$ and

1

$1$ .

\frac{36 + 4 i}{82}

$\frac{36 + 4 i}{82}$

\frac{36 + 4 i}{82}

$\frac{36 + 4 i}{82}$

\frac{36 + 4 i}{82}

$\frac{36 + 4 i}{82}$

Step 3

Cancel the common factor of

36 + 4 i

$36 + 4 i$ and

82

$82$ .

Tap for more steps...

Step 3.1

Factor

2

$2$ out of

36

$36$ .

\frac{2 (18) + 4 i}{82}

$\frac{2 (18) + 4 i}{82}$

Step 3.2

Factor

2

$2$ out of

4 i

$4 i$ .

\frac{2 (18) + 2 (2 i)}{82}

$\frac{2 (18) + 2 (2 i)}{82}$

Step 3.3

Factor

2

$2$ out of

2 (18) + 2 (2 i)

$2 (18) + 2 (2 i)$ .

\frac{2 (18 + 2 i)}{82}

$\frac{2 (18 + 2 i)}{82}$

Step 3.4

Cancel the common factors.

Tap for more steps...

Step 3.4.1

Factor

2

$2$ out of

82

$82$ .

\frac{2 (18 + 2 i)}{2 \cdot 41}

$\frac{2 (18 + 2 i)}{2 \cdot 41}$

Step 3.4.2

Cancel the common factor.

$\frac{2 (18 + 2 i)}{2 \cdot 41}$

Step 3.4.3

Rewrite the expression.

$\frac{18 + 2 i}{41}$

Step 4

Split the fraction

$\frac{18 + 2 i}{41}$ into two fractions.

$\frac{18}{41} + \frac{2 i}{41}$