Simplify 6/(9+i)

$\frac{6}{9 + i}$

Step 1

Multiply the numerator and denominator of $\frac{6}{9 + 1 i}$ by the conjugate of $9 + 1 i$ to make the denominator real.

$\frac{6}{9 + 1 i} \cdot \frac{9 - i}{9 - i}$

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $6$ by $9$ .

$\frac{54 + 6 (- i)}{(9 + 1 i) (9 - i)}$

Multiply $- 1$ by $6$ .

$\frac{54 - 6 i}{(9 + 1 i) (9 - i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{54 - 6 i}{9 (9 - i) + 1 i (9 - i)}$

Apply the distributive property.

$\frac{54 - 6 i}{9 \cdot 9 + 9 (- i) + 1 i (9 - i)}$

Apply the distributive property.

$\frac{54 - 6 i}{9 \cdot 9 + 9 (- i) + 1 i \cdot 9 + 1 i (- i)}$

Simplify.

Multiply $9$ by $9$ .

$\frac{54 - 6 i}{81 + 9 (- i) + 1 i \cdot 9 + 1 i (- i)}$

Multiply $- 1$ by $9$ .

$\frac{54 - 6 i}{81 - 9 i + 1 i \cdot 9 + 1 i (- i)}$

Multiply $9$ by $1$ .

$\frac{54 - 6 i}{81 - 9 i + 9 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{54 - 6 i}{81 - 9 i + 9 i - i i}$

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

$\frac{54 - 6 i}{81 - 9 i + 9 i - (i^{1} i)}$

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

$\frac{54 - 6 i}{81 - 9 i + 9 i - (i^{1} i^{1})}$

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

$\frac{54 - 6 i}{81 - 9 i + 9 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{54 - 6 i}{81 - 9 i + 9 i - i^{2}}$

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

$\frac{54 - 6 i}{81 + 0 - i^{2}}$

Add $81$ and $0$ .

$\frac{54 - 6 i}{81 - i^{2}}$

Simplify each term.

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

$\frac{54 - 6 i}{81 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{54 - 6 i}{81 + 1}$

Add $81$ and $1$ .

$\frac{54 - 6 i}{82}$

Step 3

Cancel the common factor of $54 - 6 i$ and $82$ .

Factor $2$ out of $54$ .

$\frac{2 (27) - 6 i}{82}$

Factor $2$ out of $- 6 i$ .

$\frac{2 (27) + 2 (- 3 i)}{82}$

Factor $2$ out of $2 (27) + 2 (- 3 i)$ .

$\frac{2 (27 - 3 i)}{82}$

Cancel the common factors.

Factor $2$ out of $82$ .

$\frac{2 (27 - 3 i)}{2 \cdot 41}$

Cancel the common factor.

$\frac{2 (27 - 3 i)}{2 \cdot 41}$

Rewrite the expression.

$\frac{27 - 3 i}{41}$

Step 4

Split the fraction $\frac{27 - 3 i}{41}$ into two fractions.

$\frac{27}{41} + \frac{- 3 i}{41}$

Step 5

Move the negative in front of the fraction.

$\frac{27}{41} - \frac{3 i}{41}$

Simplify 6/(9+i)

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