Write in Standard Form (4+9i)/(9-i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Expand $(4 + 9 i) (9 + i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $9$ .

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

Multiply $9$ by $9$ .

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

Multiply $9 i i$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $9$ by $- 1$ .

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

Subtract $9$ from $36$ .

$\frac{27 + 4 i + 81 i}{(9 - i) (9 + i)}$

Add $4 i$ and $81 i$ .

$\frac{27 + 85 i}{(9 - i) (9 + i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{27 + 85 i}{9 (9 + i) - i (9 + i)}$

Apply the distributive property.

$\frac{27 + 85 i}{9 \cdot 9 + 9 i - i (9 + i)}$

Apply the distributive property.

$\frac{27 + 85 i}{9 \cdot 9 + 9 i - i \cdot 9 - i i}$

Simplify.

Multiply $9$ by $9$ .

$\frac{27 + 85 i}{81 + 9 i - i \cdot 9 - i i}$

Multiply $9$ by $- 1$ .

$\frac{27 + 85 i}{81 + 9 i - 9 i - i i}$

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

$\frac{27 + 85 i}{81 + 9 i - 9 i - (i^{1} i)}$

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

$\frac{27 + 85 i}{81 + 9 i - 9 i - (i^{1} i^{1})}$

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

$\frac{27 + 85 i}{81 + 9 i - 9 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{27 + 85 i}{81 + 9 i - 9 i - i^{2}}$

Subtract $9 i$ from $9 i$ .

$\frac{27 + 85 i}{81 + 0 - i^{2}}$

Add $81$ and $0$ .

$\frac{27 + 85 i}{81 - i^{2}}$

Simplify each term.

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

$\frac{27 + 85 i}{81 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{27 + 85 i}{81 + 1}$

Add $81$ and $1$ .

$\frac{27 + 85 i}{82}$

Step 3

Split the fraction $\frac{27 + 85 i}{82}$ into two fractions.

$\frac{27}{82} + \frac{85 i}{82}$

Write in Standard Form (4+9i)/(9-i)

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