Algebra Examples

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

Step 1

Multiply the numerator and denominator of $\frac{4}{9 + 1 i}$ by the conjugate of $9 + 1 i$ to make the denominator real.

$\frac{4}{9 + 1 i} \cdot \frac{9 - i}{9 - i}$

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

$\frac{4 \cdot 9 + 4 (- i)}{(9 + 1 i) (9 - i)}$

Step 2.2.2

Multiply $4$ by $9$ .

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

Step 2.2.3

Multiply $- 1$ by $4$ .

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(9 + 1 i) (9 - i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

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

Step 2.3.1.2

Apply the distributive property.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $9$ by $9$ .

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

Step 2.3.2.2

Multiply $- 1$ by $9$ .

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

Step 2.3.2.3

Multiply $9$ by $1$ .

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

Step 2.3.2.4

Multiply $- 1$ by $1$ .

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

Step 2.3.2.5

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

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

Step 2.3.2.6

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

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

Step 2.3.2.7

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

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

Step 2.3.2.8

Add $1$ and $1$ .

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

Step 2.3.2.9

Add $- 9 i$ and $9 i$ .

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

Step 2.3.2.10

Add $81$ and $0$ .

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

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

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

Step 2.3.4

Add $81$ and $1$ .

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

Step 3

Cancel the common factor of $36 - 4 i$ and $82$ .

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Step 3.1

Factor $2$ out of $36$ .

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

Step 3.2

Factor $2$ out of $- 4 i$ .

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

Step 3.3

Factor $2$ out of $2 (18) + 2 (- 2 i)$ .

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

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $2$ out of $82$ .

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

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.

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

Step 5

Move the negative in front of the fraction.

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