Write in Standard Form (9+i)/(9-i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $9$ by $9$ .

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

Move $9$ to the left of $i$ .

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

Multiply $i i$ .

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

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

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

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

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

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

Add $1$ and $1$ .

$\frac{81 + 9 i + 9 i + i^{2}}{(9 - i) (9 + i)}$

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

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

Subtract $1$ from $81$ .

$\frac{80 + 9 i + 9 i}{(9 - i) (9 + i)}$

Add $9 i$ and $9 i$ .

$\frac{80 + 18 i}{(9 - i) (9 + i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{80 + 18 i}{9 (9 + i) - i (9 + i)}$

Apply the distributive property.

$\frac{80 + 18 i}{9 \cdot 9 + 9 i - i (9 + i)}$

Apply the distributive property.

$\frac{80 + 18 i}{9 \cdot 9 + 9 i - i \cdot 9 - i i}$

Simplify.

Multiply $9$ by $9$ .

$\frac{80 + 18 i}{81 + 9 i - i \cdot 9 - i i}$

Multiply $9$ by $- 1$ .

$\frac{80 + 18 i}{81 + 9 i - 9 i - i i}$

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

$\frac{80 + 18 i}{81 + 9 i - 9 i - (i^{1} i)}$

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

$\frac{80 + 18 i}{81 + 9 i - 9 i - (i^{1} i^{1})}$

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

$\frac{80 + 18 i}{81 + 9 i - 9 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{80 + 18 i}{81 + 9 i - 9 i - i^{2}}$

Subtract $9 i$ from $9 i$ .

$\frac{80 + 18 i}{81 + 0 - i^{2}}$

Add $81$ and $0$ .

$\frac{80 + 18 i}{81 - i^{2}}$

Simplify each term.

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

$\frac{80 + 18 i}{81 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{80 + 18 i}{81 + 1}$

Add $81$ and $1$ .

$\frac{80 + 18 i}{82}$

Step 3

Cancel the common factor of $80 + 18 i$ and $82$ .

Factor $2$ out of $80$ .

$\frac{2 (40) + 18 i}{82}$

Factor $2$ out of $18 i$ .

$\frac{2 (40) + 2 (9 i)}{82}$

Factor $2$ out of $2 (40) + 2 (9 i)$ .

$\frac{2 (40 + 9 i)}{82}$

Cancel the common factors.

Factor $2$ out of $82$ .

$\frac{2 (40 + 9 i)}{2 \cdot 41}$

Cancel the common factor.

$\frac{2 (40 + 9 i)}{2 \cdot 41}$

Rewrite the expression.

$\frac{40 + 9 i}{41}$

Step 4

Split the fraction $\frac{40 + 9 i}{41}$ into two fractions.

$\frac{40}{41} + \frac{9 i}{41}$

Write in Standard Form (9+i)/(9-i)

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